Depends on if paper is available.
For undergraduate level mathematics at least, it is fairly common a lecturer will produce their own online notes for a course rather than assigning a textbook. This is essentially writing a small book.
If paper is available, your lecturers should be required to produce such books. Then each student, as part of their degree work, is required to make manual copies of several books.
In fact you can pay scribes who are not students. They don't even have to understand what they're copying. Remember the apocalypse was only a few decades ago so literacy is much more common in your world than during the dark ages (for now).
Soon you have a respectable library.
If paper is unavailable then even students 'taking notes' will be impossible. This is an even bigger obstacle than the nonexistence of a printing press and warrants several questions of its own.
Bonus: I don't know how much easier it is to build a paper factory than a printing press. But the university should have its own paper factory and students are required to put in hours to manufacture all the paper they need, or pay extra fees for the labour.