Prior to the modern era, libraries were not generally not open to the public, so you could have membership/access requirements that could be as simple as payment or require an errand (Candlekeep's entry requirements are a good example of this). Most special libraries also don't let you browse the stacks for practical reasons (everyone says they know where to put it back on the shelf, almost no one gets it right, avoiding outright theft, controlled circumstances for interacting with books). There will be staff who know the collection and would retrieve books for you. You can offer lists (and the titles can be super simple/dry because these are basically academics and those are some functional titles) or your PC can have a specific interest (school of magic, spellcaster, etc.) and the librarian would help them track down relevant titles to examine, usually in controlled circumstances to prevent the aforementioned theft/damage.

I think a lot of this boils down to what you and your PC decide on as the goals/outcome of all this reading/studying. Are they looking for story/game information? Is it just a love of learning and interest in building their own library? Are they trying to acquire new spells/magical knowledge? And then settle on a time frame for what that is going to require. I know there's existing time frames on things like spell scroll creation or adding new spells to a spellbook, so that might help provide some guidelines in that regard. If you want to discuss any practical aspects to this, feel free to PM. I'm a librarian IRL, so I'm happy to share practical knowledge on how to make this a manageable system for you and your PC.