Largely he doesn't.
His entry through Admissions he gets through on Ben's teaching, nerve, and a healthy splash of cheating. His admission to the Arcanum, whipping, and subsequent banning from Archives is less than a span later; he spent less than two hours inside, most of that studying one curious bestiary. His second time through, he got help from friends:
Despite the long days I spent studying with Wil and Sim, admissions were difficult. I breezed through many of the questions, but Hemme was openly hostile, ... Chapter 49
After that, he pushes exhaustingly through his second term, and by the time he gets to the end of term and the next Admissions (right after his performance at the Eolian, which gave him some room to breathe moneywise)...
But as I turned back to the Mews, a thought occurred to me. How much preparation did I really need? More importantly, how much could I genuinely accomplish without access to the Archives?
Thinking it over, I raised my hand over my head with my middle finger and thumb extended, signaling that I had a slot five days from now that I was willing to sell. Chapter 60
After that point, he starts trading his slot for cash or favors, and his trade is brought up every time he has to go back through Admissions. He can't really study without access to the Archives, has to rely on his memory and talent (which he is lucky to have in spades) with whatever he has picked up in class, and the two best friends he relies on he is quickly outstripping, not to mention he wants them more as friends than as study crutches. There is only one other exception...
... when he is drugged right before his exam, and Fela gives up her slot to give him time to recover.
The Archives are pretty much it, as far as libraries go, aside from the Masters lectures and courses. If he had any real way to study any adequate source of knowledge, he wouldn't have spent so much time trying to find secret ways into the Archives.
You may also note that after his second-term burnout, and skipping most of the intro courses (which is where Hemme got to be vindictive with those essays), most of his courses he sticks to are practical and hands-on knowledge stuff - Artificing, Advanced Sympathy, and Medica. Also all of these are courses where a handful of friendly faces are present to help him out on occasion (Manet, Sovoy, and Mola, respectively).