This, Shmoopers, is one of the few occasions where we feel comfortable saying that the speaker of this poem is Ben Jonson, the author. Your English teachers have cautioned you against making this assumption, we're sure, but Shmoop has got some seriously convincing supporting evidence for our case.
Consider the following:
(1) This poem was first published in 1623 at the beginning of Shakespeare's First Folio. The practice of including elegies and other commendatory poems written by friends at the beginning of a deceased author's work was very common, but this front section was sacred for just that reason; it provided a unique space for other authors to share their personal thoughts, feelings, and reflections on the life and work of a peer. Jonson's poem, if not written from a first person perspective, would have been an odd thing for the editors to include.
(2) Enough of Jonson's personal writings have survived that we know a little bit about how the man's mind worked, and this poem screams Jonson. Obviously the style and rhetoric are his, but the sentiments expressed by the speaker are so quintessentially Jonsonian it practically smacks you in the face. The little dig about "small Latin and less Greek"—totally in line with Jonson, the hypercritical classicist. And the comments about Shakespeare's "natural" gift (as opposed to the very intentional labor that Jonson put into his poetry) are perfectly in line with opinions of Shakespeare's work Jonson had expressed before he died.
(3) The poem shows off the intelligence of the author in a very Jonsonian way. It spends a lot of time talking about how Shakespeare is better than the greats of Greece, Rome, and England, but veiled in that praise is Jonson showing off his own knowledge. His learning was one of the things that set him apart from Shakespeare, and no way would Jonson strut his stuff without taking the credit.
(4) Shmoop would never lie to you. Ever. And we say the speaker is Jonson.