The bottom line is that both answers are correct, and the textbook or teacher claiming that only one is correct is wrong. Most can be a superlative, and if it's meant that way (if the writer thinks that this is the very best time of one's life for discovery and learning), then the is the answer. But most, as Loob points out, can also mean "extremely", and if that's the meaning intended (it's a very good time, not necessarily the best), then a is the answer. I suspect that the latter was probably intended, but that suspicion doesn't make "the" wrong. The sentence is ambiguous.
I agree with E2efour and Loob that "university" is not a time but a place. In AmE, we'd likely say "The college years are . . . " (we use "college" as a more general word, whether or not the college attended is part of a university).