Ulysses is the Latin form of the Greek Odysseus, stemming from the Sicilian or alternate Latin form Ulixes. The first instance of these forms in literature that I can find is in the _Odusia_ by Livius Andronicus. This is an early translation of the Odyssey (third century BC). The only parts of it that survive are 46 lines from 17 books of the Odyssey, but this is one of those lines:
> Inferus an superus tibi fert deus funera, Ulixes? (Od. 10, 64)
I found the surviving text of the Odusia here.
Marcus Pacuvius was an early Latin dramatist who probably used Ulysses. Though I'm not sure, he wrote many of his plays on Greek topics, and the translations of his works I found use the name Ulysses.
So to answer your question: **Livius Andronicus** was the first writer I can find who used a form of the Latin name of Odysseus, and Virgil was not the first.