To sound the depth of that thou wilt profess:

The end of the debate is that he will deepen his knowledge of the art of magic and summon demons to do his bidding and give him the unlimited power he covets. Thus he will add one more area of proficiency to his breadth of knowledge.

A sound magician is a mighty god:
Here, Faustus, tire
thy brains to gain a deity.

From the descriptions above, it is clear that Doctor Faustus does indeed fit both the concept of a man of learning living during the 14th and 15th centuries as well as fit the modern construct of the Renaissance man.

He was a lover of arts, a connoiseur of many disciplines of all forms, a practicing scholar and a philosopher, a dweller in the supernatural, an orator, a professor, and a student of a myriad of different fields. He was able to fit in every circle, even though his own petulance madfe him unable to be normal. Yet, these elements made him whatwe call these days a "Renaissance Man," or a man who can basically do it all with grace.

Dr. Faustus has aspirations of acquiring knowledge that reach far above the commonly perceived boundaries of the time - maybe even taking him beyond the 'forbidden tree of knowledge'But he doesn't necessarily have the skills to attain this unearthly intelligence, or the linguistic prowess or imaginative scope. The potential merits of existing or even prospering in a world without a God seem unattainable - we and Faustus are only human. However, we can identify with him as we are all questioning curious beings and there are answers that we would all like to have whether they be about the limits of the universe or the cause of the Big Bang. Whether we would be prepared to go beyond the bounds of decency or moral limits is where the difference lies. Also, would we use the privilege of unearthly knowledge to do good or wise acts, or seek to further our own trivial materialistic needs? Faustus seems a Renaissance figure in his infinite appetites and the things he wants and also in his willingness to rid his world of everything that stands in the way of his own illusory enjoyment. Faustus very nearly "had it all" but wouldn't have deserved it and threw it all away.

A common definition of the term "Renaissance man" is someone who is accomplished in many areas of his life, including the arts, learning, and physical pursuits. Another way to define the term is for someone to be a man of the Renaissance; that is, being a man whose thinking and actions are typical of Renaissance actions and thinking. Using the latter definition, Doctor Faustus is, indeed, a Renaissance man in Doctor Faustus by Christopher Marlowe.

One of the primary differences in thinking between the early Renaissance man and a Medieval man was how he sees religion. While Medieval thinking was God-centered, Renaissance thinking was man-centered. The Renaissance man spent his time pursuing knowledge and exploration of all kinds, particularly scientific exploration, with little regard for God or religion. That is exactly how Doctor Faustus lives his life, in the pursuit of of knowledge, wealth, power, and the mystical arts.

In his first speech, Doctor Faustus outlines the key areas of learning, citing Aristotle (for logic), Galen (for medicine), Justinian (for law), and the Bible (for religion). These were fine for another time (the Medieval period), but he rejects these as worthless notions for his world (Renaissance England). Instead he dismisses God and says he wants to pursue his own desires: magic and the search for things that matter to him, such as riches, knowledge, and power. 

What doctrine call you this: Che sera, sera
What will be, shall be? Divinity, adieu.
These metaphysics of magicians
And necromantic books are heavenly;
Lines, circles, letters, characters.
Ay, these are those that Faustus most desires.
O what a world of profit and delight,
Of power, of honour, and omnipotence
Is promised to the studious artisan?
All things that move between the quiet poles
Shall be at my command. Emperors and Kings,
Are but obeyed in their several provinces,
But his dominion that exceeds in this,
Stretcheth as far as doth the mind of man:
A sound magician is a demi-god.

It may seem ironic, then, that the primary conflict of this story is the conflict between God's decrees (of everlasting life and eternal damnation) and man's quest to be god, or at least a god. 

Marlowe seems to be warning his audience that taking this new way of thinking (the pursuit of personal glory, knowledge, and riches) apart from spiritual concerns is destined to end badly. Doctor Faustus gladly agrees to sell his soul for these things and he gets them; however, his life degenerates into cruelty and unhappiness. He does have opportunities to make different choices (repent) but does not; Mephistopheles comes for him and Doctor Faustus forfeits his life for things that mean nothing in the end.

As a Renaissance man who represents the thinking of his time (the Renaissance), Doctor Faustus does fit this description. 

Your question isn't exactly clear. You are employing an English language idiom "pay the price" that does not have a literal meaning in this usage. What the idiom means is that you suffer the negative consequences or results of something wrong or risky that you or someone has done:

• McGraw-Hill Idiom Dictionary: to suffer the consequences for doing something or risking something

In light of this idiom definition, your question is really asking, in paraphrase:

• Is Doctor Faustus a Renaissance Man who had to suffer unpleasant consequences for his actions that were more appropriate for a Medieval Man to have suffered?

Answering this requires understanding three things: (1) the primary relevant characteristic of a Renaissance Man; (2) what wrong or risky thing Faustus did that he'd have to "pay the price" for; (3) the contrasting primary and relevant characteristic of a Medieval Man.

(1) Faustus exemplified the Renaissance celebration of the expansion of knowledge. The rediscovery of ancient Greek and Latin texts opened new fields of study and new ideas for Europeans and excited the minds of the intellectuals leading to a blossoming of intellectual and scientific thought (this later was labeled the Renaissance). This exploration of knowledge put a new light on individual accomplishment and individual identity because intellectuals could see what others had done before them and what being in the shadow of a religion-dominated intellectualism had left unexplored.

(2) Taking the Renaissance spirit of exploration and individual freedom and power to the extreme, Faustus wanted unlimited knowledge (knowledge of God and his ways) and unlimited power (for wealth and control of the natural elements).

FAUSTUS. O, what a world of profit and delight,
     Of power, of honour, of omnipotence,
     ...
     All things that move between the quiet poles
     Shall be at my command:
     ...
     A sound magician is a mighty god:

(3) Contrastingly, Medieval Man was focused on the doctrine of the Christian Catholic Church and was governed by the presence, if not the law (though sometimes both), of the Church. Best known among the great Medieval thinkers were Charlemagne and Thomas Aquinas. Charlemagne was integral is establishing the Holy Roman Empire and Aquinas was integral in detailing Church doctrinal interpretations. Because religion so permeated Medieval life and thought (think of the religious themes and satire in The Canterbury Tales), good and evil, damnation and salvation, forgiveness and punishment were dualistic ideas that held dominance over even common people, like those peasants who bought Indulgences from pardoners. 

Now to your question: Faustus pursued the Renaissance ideals in knowledge and individualism. In this way he is the ideal Renaissance Man. Yet, he pursued these to their extreme limits: he sold his soul to Lucifer via Mephistopheles. In this way he was thrust into and re-enlivened the narrow emphases of Medieval thought. This may be seen as a Renaissance Man suffering the consequences of a Medieval Man's thought system: he was thrust into the living dualities of good and evil, damnation and salvation, forgiveness and punishment and was subjected to the domination of Lucifer (ironic because Faustus sought complete domination of the universe). In this sense, the answer is "Yes," though it is also "No" because the principles of Christianity had not died in the Renaissance, they had only been reduced in intensity.

Perhaps the greatest statement of Renaissance humanism occurs in Pico della Mirandola's "Oration on the Dignity of Man." The Neo-Platonic Pico emphasized human achievement and the importance of human's pursuit of knowledge. The Oration claims that through the exercise of the intellect, humans can ascend or descend the chain of being, and that this intellectual capacity, coupled with free will, makes humanity the most wondrous of creatures. Transferred to more tangible circumstances, Pico's "dignified man" can become Marlowe's "over reacher."

At the beginning of Marlowe's play, we see that Dr. Faustus is a person of middling birth who has improved himself though studies. He is superior in philosophy, law, and medicine. He seems to have "plumbed the depth" of human learning and seeks more. He has accomplished marvels in human terms and seeks to ascend to a level comparable to the metaphysical. This approach to human intelligence and the longing for more is a quintessential Renaissance desire. Even when we consider what necromancy offers him, we find that what he ends up accomplishing through Mephistopheles' help is exactly what the Renaissance was already offering (cheap theatrical tricks in conjuring ancient figures, getting food out of season by traveling to another part of the earth, pranking the Pope and undercutting his dignity).

Of course, Faustus fails to understand his reading of the Bible in the first scene, largely because he picks and chooses passages too carelessly, and thus fails to see the coherence in divine writ. In the end, he suffers the tragedy of a medieval man, and the medieval "psychomachia" or soul war represented by the good and bad angel, confirms that his end will be as grim as a medieval sinner would expect.

Doctor Faustus is a Renaissance hero in that he wants to develop his abilities to their fullest extent. Possessed with a firm belief in man's almost limitless potential, Faustus wants to push back the boundaries of what human beings are capable of, especially in relation to the acquisition of knowledge.

Yet Faustus's seemingly boundless confidence soon turns to despair as he realizes his own—and by extension, man's—limitations. Like all human beings, his reach greatly exceeds his grasp; he wants something he ultimately cannot have. Faustus's despair leads him to conclude a diabolical bargain with Lucifer, to whom he agrees to sell his soul; this is the only way he can achieve the kind of knowledge—and the power that goes with it—that he so desperately craves.

Faustus may be a Renaissance hero, but he also stands as a salutary warning against the kind of intellectual hubris to which the rediscovery of ancient learning frequently led.

The title character of Christopher Marlowe's play Doctor Faustus has often been called a "Renaissance man," and the designation seems appropriate in number of different ways.  Those ways include the following:

• He is, quite literally, a man living during the historical period commonly thought of as "the Renaissance" (that is, the sixteenth century, at least in England).
• He exemplifes the strong interest in classical literature and culture that was typical of that period.
• At the same time, he shows the influence of Christian ideas, which were also extremely important during that era.
• He exemplifies the greater social mobility common during that period. His parents were "base of stock" (Prologue, 10), but their son nevertheless rose to great social prominence and power.
• He exemplifies the growing importance of formal education and especially university training that was common during this time. He has been to a university and is extremely well educated (which makes his later foolish choices all the more difficult to understand).
• He is interested in, and even expert in, many different areas of learning -- a sense in which we still use the phrase "Renaissance man" today.
• He has the kind of high-flying aspirations that were typical of many people during the Renaissance -- a period (for instance) of enormous geographical exploration.
• He is a "Faustian man" in Oswald Spengler's sense: a man always striving for another achievement, never content with what he already has (although the achievements in this case seem ultimately trivial).
• His chief focus seems to be earthly rather than heavenly, although the main purpose of the play seems to be to check and warn against precisely the kinds of materialistic impulses that motivate Faustus.
• He is an "over-reacher," as was true of many people in the Renaissance. Therefore, almost the final words of the play warn us to

. . . Regard his hellish fall,
Whose fiendful fortune may exhort the wise
Only to wonder at [that is, merely regard with wonder, not actually perform] unlawful things.  (Epilogue, 4-6)