Ha! The ancient world was full of Greek language teachers and books about Greek grammar! More, it was full of teachers and books about Greek rhetoric - ie, the appropriate use of Greek figures of speech, as well as how to compose and deliver both oratory and written works. You were supposed to study Homer and the other epics, the classical Athenian tragedies, speeches by guys like Demosthenes, contemporary great writers… on and on. Every young Roman man whose dad thought he needed polish was sent to Athens to learn Greek rhetoric in Greek; and most rich Romans in Paul’s time had Greek slaves (paedagogues) to teach their kids Greek from early childhood. (Sorta like English kids with French governesses, except for the whole slave part.)
So yes, if Paul wanted to study, he surely could.
However… why would he, except for polish? (And he probably did take a bit of rhetoric, because he does use rhetorical figures like an educated man.)
Paul grew up in Tarsus!! It was a Greek-speaking town!! The man spoke Greek like a native (or at least Asia-Minor-style Greek), because he was a native speaker! If anything, Paul’s Greek is difficult to follow because he makes it do things that only a native speaker would make it do.
(Aramaic and Hebrew would have been only his second and third languages, unless his parents spoke Aramaic at home.)
So yes, Paul knows exactly what he’s saying when he is saying it in Greek. He understands the grammatical implications of his words a lot better than a native English speaker understand the implications of English, because that sort of thing was a focus of both classical and rabbinical education - whereas we moderns tend to ignore it.
