I don't know much about phrasal verbs (they hadn't yet been invented when I was in school), but I take "graduated" as a simple verb in the simple past tense, and "with first-class honors" as a prepositional phrase, modifying the verb. I'd call "first-class" an adjective modifying the noun "honors," which functions as object of the preposition, "with."
To me, the rest is okay.