Pride, also called hubris, is the cause of Oedipus's downfall. On a human level, it is pride that causes Oedipus to unknowingly kill his father, Laius, on the road to Thebes. His sense of pride—his sense that Laius should move aside for him and not vice versa—drives Oedipus to get angry enough to murder the man who will not let him pass. Much later, as the play opens and Creon returns from the oracle with the news that someone's sin has caused the plague in Thebes, Oedipus has too much pride (hubris) to imagine it could possibly be him.

This human pride is connected to a much more fatal form of pride, according to the Greek worldview. Oedipus has the pride to think he can beat what the gods have foreordained. That is a terrible sin. When he learns, while living in Corinth with the people he thinks are his parents, that he was fated at birth to murder his father and marry his mother, he believes he can make his own fate. He heads out from Corinth, believing that he has escaped from the will of the gods.

Oedipus doesn't learn wisdom until he is able to accept that whatever the gods prophecy is what will come to pass. When he realizes that he has, in fact, murdered his father and married his mother, he is so overcome he stabs his eyes out with pins—but at that point gains true insight and humility.

Oedipus is genuinely concerned by the damage the plague is doing to his people and seeks to help. Creon informs him that the plague is the result of King Laius's murder and that the murderer must be found and killed or expelled. Oedipus seeks counsel from Teiresias the prophet. However, the prophet is afraid of divulging the truth about the situation. He knows that Oedipus unknowingly killed the king, his own father, and married his own mother, making him directly responsible for the plague. 

Teiresias tries to dissuade Oedipus from seeking the truth, which is clearly more than he can bear, but Oedipus goads Teiresias into talking. He is angered at what he discovers, but his anger is directed at Teiresias and Creon because he believes they are reaching for his throne. Creon tells Oedipus that his temper is his worst enemy.

I'll go, and they have known my innocence. Your temper is your own worst enemy

However, the truth is still revealed to Oedipus by credible sources. Although Oedipus' fate is already sealed at birth, some suggest that if he had not killed his father then tragedy would not have befallen him. Oedipus's actions are fueled by anger and although he had a right to feel aggrieved when the king's servant forcefully pushed him out of the way, Oedipus overreacted by killing almost all of the men out of anger. 

The one who led the way, and the old man himself, wanted to push me out of the road by force. I became angry and struck the coachman who was pushing me.

Oedipus suffers from deep-seated hubris that causes him to overreact out of anger. Thus, his anger is preceded by his pride, making pride his strongest flaw.

I agree with the other commenter that it is, ultimately, Oedipus's pride that is responsible for his downfall.  Anger is a tempting answer because Oedipus does get so angry at Teiresias, at his brother in law, Creon, even at Jocasta, his wife and mother.  However, Oedipus's anger, if we trace it, stems from his pride.  He originally kills Laius on the road, in anger.  But why was he angry?  Because Laius and his retinue attacked Oedipus, insulting his pride.

Why does he get so angry at Teiresias, the prophet?  Because Teiresias claims that Oedipus doesn't really want to know the truth, that it would be better for Oedipus to remain in the dark.  When Teiresias insists that he knows better than Oedipus what is best, he wounds the king's pride.  Oedipus then assumes that Teiresias must be working with Creon in order to take his crown.  The idea that his brother in law believes that he is not a sound or deserving ruler (totally made up in Oedipus's own head) wounds his pride, and he becomes angry.

Oedipus makes some errors in judgment as a result of his tragic flaw, pride, and one of these errors is his quickness to become angry when someone says something he doesn't like.

The downfall of the protagonist of a tragedy is usually caused by the character's "tragic flaw". This is true of Oedipus as much as it is true of any other tragic figure of antiquity. Of your two choices, pride or anger, it is pride that comes nearest to identifying the tragic flaw of this character. 

The ultimate cause of Oedipus' downfall is his unwillingness to accept his fate. He cannot accept the predictions about his life (that he will murder his father and sleep with his mother) and he fights against them.

Oedipus struggles against the oracle that predicts his hand in his father's death...

To do so, Oedipus must reject the very idea of fate and the idea that the gods are in control of the lives of mortals. This rejection can be seen as evidence of his great pride. 

Additionally, Oedipus invites information, however damaging it might be, saying that he can handle any truth that comes his way. This also suggests a vanity and pride on his part and, ultimately, leads to his tragedy. Though his wife and others try to convince him to stop his investigation, Oedipus continues and is destroyed by his discovery:

by demanding that others tell him all they know he is forced to confront the hideous facts of his patricide and incest.

Does Oedipus resolve his own dilemma of having too much pride in Sophocles's work Oedipus Rex?

This is an excellent question because it gets to the heart of the story. Oedipus is challenged throughout the play to confront his own pride and humble himself to realize the evil he has done. His own pride and haughtiness have caused him to murder his own father and marry his mother, while reveling in the praise of the people—essentially building himself up before his own ultimate downfall.

The poetic element occurs when the Blind Prophet Tiberias tells him that he is blind to his own evil, at which Oedipus scoffs. In the end, to emphasize the point of Oedipus's realization, he gouges his own eyes out in shock and revulsion at what he has done. The irony is that, although he is now physically blind, he is spiritually aware of his deeds. Because of this, it seems clear that he has confronted his pride and humbled himself.

The hubris that brought him to challenge fate itself and eventually fall into a trap, becoming the very thing he sought to flee from, is confronted by his realization that he is not the great and mighty conqueror he believes himself to be. Instead, he realizes that he is foolish and full of violence and pride. Finally, he learns his mistake, but is permanently scarred by his own actions.

Does Oedipus resolve his own dilemma of having too much pride in Sophocles's work Oedipus Rex?

The moral of the story Oedipus Rex, the epic play of antiquity, is not to be overcome by pride like the titular hero. In the tale, Oedipus believes he can overcome fate when the Oracle tells him that he will murder his father and marry his mother. After inadvertently causing this to happen, he becomes a hero and a king. Unfortunately, he eventually is ruined by the knowledge of what he has done. The blind prophet Tiberias warns him of his folly and tells Oedipus that he is truly the blind one.

Oedipus eventually comes to the realization that he had fallen into fate's trap and committed the actions he was foretold to. At this point, he has to come to grips with reality, and in this moment, he is forced to deal with his pride. He laments his actions and realizes that he was not the great conquering man he believed himself to be but was instead a murderer who had committed incest. He also realizes in this moment that he had not been able to outsmart fate itself. This realization causes him to humble himself, removing his pride.

In the end, Oedipus's eyes are opened to his folly. When he knows the truth, he is no longer proud, but becomes a humble king.

Does Oedipus resolve his own dilemma of having too much pride in Sophocles's work Oedipus Rex?

Oedipus' pride is what keeps him comfortably in denial over the course of the play. When he is in conflict with Tiresias, who knows the truth, his pride allows him to simply believe that the blind seer is in league with Creon and attempting to overthrow him. However, when Jocasta says that Laius was killed at a place where three roads meet, Oedipus realizes that he has killed men in a similar setting.

It could be said that Oedipus resolves his pride at the end of the play. He certainly no longer seems consumed or even compelled by pride when he blinds himself with a dead Jocasta's broach. He demands that Creon exile him from Thebes forever. He has reached such a level of shame that, even as a blind man, he does not care to be in the presence of anything familiar. For better or for worse, his pride has been dealt with.

Does Oedipus resolve his own dilemma of having too much pride in Sophocles's work Oedipus Rex?

One could say that Oedipus does indeed resolve the dilemma of dealing with his overweening pride, but only by a drastic act of self-mutilation. By the time Oedipus gouges out his own eyes, the damage has already been done. Though rather extreme, to say the least, his actions are far too little, far too late. And even after this particularly graphic and unpleasant scene, it's still not entirely clear that Oedipus has managed to get a grip on his hubris. For instance, in Sophocles's later play Oedipus at Colonus, the now-blind Oedipus shows great arrogance and pride by remaining seated in an olive grove even after he's been told that it's sacred to the Furies.

Once again, Oedipus shows that he's prepared for other people to incur the wrath of the gods for his actions, just as he did when he brought plague upon Thebes in Oedipus Rex. If we look solely at Oedipus Rex, then it seems that Oedipus has indeed resolved his dilemma. But if we take Oedipus at Colonus into consideration, then we can't say the same thing with any degree of confidence.

Does Oedipus resolve his own dilemma of having too much pride in Sophocles's work Oedipus Rex?

This is a great question. Here are a few points to consider. 

First, as you state the tragic flaw of Oedipus is his pride. In fact, I would go as far as to say that the whole play is driven by Oedipus' pride. In light of this, it might seem that Oedipus does not resolve his pride issue. However, there are a few indications to show that Oedipus finally does realize his pride.

First, by the end of the play, Oedipus definitively knows that he is the one who committed the crimes. In light of this, he is forced to deal with his pride. There is simply nothing that he can do. More importantly, he accepts his fate. We can see this even more clearly in the two other Theban plays. In the final play, he becomes a blessing for Athens. He is now a wise man. 

Second, there is a motif in the play of blindness. In the beginning of the play he see physically, but he cannot see figuratively. We can say that he is blind to his pride. By the end, he blinds himself physically, but he is now able to see much more on a spiritual level. This suggests that he does deal with his problem of pride. 

In Oedipus Rex, is Oedipus brought down by arrogance or pride?

According to Aristotle's view of tragedy in his Poetics, a work which has dominated critical thinking since it was written, the protagonist's misfortunes result not from character deficiencies but from what Aristotle terms hamartia, a criminal act committed in ignorance of some material fact or even for the sake of a greater good.  The criminal act that Oedipus commits is the killing of the man who blocks his way on a road, a man who, unbeknowst to Oedipus, is his father.  This act may be motivated by arrogance as Oedipus desires to show his superiority to the other man; however, his downfall is the result of hamartia, and neither pride nor arrogance. 

This arrogance of Oedipus is evident in his thinking that he can solve the riddle of what causes the plague from which Thebes suffers.  But, at the same time, Oedipus is a good king and a sound ruler who feels confident in his problem-solving abilities since he already has solved the riddle of the Sphinx.