First, in the opening lines, Jonson establishes himself as one who is able to comment on his mentor: "To draw no envy, Shakespeare, on thy name,/Am I thus ample to thy book and fame;". Johnson himself was a proflic and popular writer. He is saying here that his works match that of Shakespeare's in both sheer page count and also in popularity.
However, Jonson quickly confesses that there can be no writer who has ever lived equal to Shakespeare's genius:
"The applause ! delight ! the wonder of our stage!
My Shakespeare rise ! I will not lodge thee by
Chaucer, or Spenser, or bid Beaumont lie
A little further, to make thee a room:"
Furthermore, his mentor has succeeded even these famous writers by equaling and surpassing even the most lauded writers ever: "Aeschylus,Euripides, and Sophocles" among them.
The theme of the poem truly can be understood in this single line: "He was not of an age, but for all time!"
Jonson, while he knows he is a good and respected writer, realizes that Shakespeare is in a class by himself and that his own talents pale by comparison.
Is this the same Ben Jonson who talks rather dismissively of the poet ape? Of WS lack of knowledge of latin and greek? To draw 'no envy' on thy name- he even gives us the required spelling in case we should use the wrong one. I suspect it may be an anagram - the resultant phrase being "ample to thy book and fame." Far from acknowledging himself as a much inferior play write I would suggest he is 'joking' to those in the know- the fawning adulation may well be sarcasm- but even just a brief read through tells us that there's a lot of wordplay and curious sentence structure that is lost on many who are not looking for it. It may also easier to appreciate if you are at least open to the possibility WS from Stratford who was also identified by Greene (or Nash) as being an 'upstart crow' (crows being used as metaphors for stealing/ plagiarism) and the reasonably argued theories that the body of work is wrongly attributed to him.