Difficult to say, because we're not really sure how widespread literacy (and hence potential demand for books) was. Though the fact that Roman monuments often had writing on them does suggest that at least a significant minority had some reading abilities. And of course, when Christianity comes around you'll have plenty of Bibles to print.
Alternatively, maybe the printing press could get going for some more niche uses where accuracy was important. E.g., if you had a monarch giving orders to one of his governors, or two kings concluding a treaty, or a law, or a business contract, copies of the relevant text could be printed out to make sure that all parties had access to the exact same wording without any copying errors. This could get the printing press in widespread use, with book-printing as a later, secondary development.