To build such a library you need the free labor beyond farming and other subsistence levels to do the construction. Then you need the industry to make paper, ink, movable type, printing presses, bindery materials, etc.
A library like the one at Alexandrea required all that too. If you plan on putting in works by world scholars, etc., in addition to stuff gained from the future, you need a means to transcribe all of it into texts.
Another problem might be the ship's computer and other electronic storage doesn't contain across-the-board knowledge that could be necessary for rapid technological advances.
As I pointed out, just making something like a musket from scratch 500 or so years early requires considerable advances in numerous technologies as well as in science.
To make a musket locally requires the technologies for mining the necessary ores, then smelting them, then making them into the proper shapes. A musket barrel requires a considerable amount of technology on it's own. This won't happen with a small population base.
So, given the nature of the Roman empire by the Imperial period, it might be better again to go with a variant of the Byzantine Empire. That is, you set up a new and separate colony where it can expand rapidly in terms of population to give you the manpower base to exploit for making technology and industry. If you carefully add in the better minds available to do things like transcribe literature into books, and learn the technology so it's transferable, you can make it a stable republic right from the get go. Let the original wither on the vine while creating a new Rome in the Americas.
Instead of enslaving the indigenous population you turn them into Romans and make them willing participants. With minimal warfare and plenty of room to expand, this empire becomes a world beater in relative isolation. They know about the rest of the world but the rest of the world doesn't know about them.
Thus, when they do decide to reestablish contact after several centuries or more, they are so far in advance of the rest of the world they can easily take control. The biggest thing your benefactors have to do is make sure they don't end up an imperial colonial monarchy or dictatorship but remain faithful to being a republic focused on world enlightenment.
This might be summed up by saying they show up bearing blue jeans, soda pop, and chocolate bars. Being centuries ahead of everybody else means they become the model for advancement but since they aren't simply taking over and forcing the locals to bend to their will means the locals elsewhere have to become capitalists and entrepreneurs to catch up.
Dispose of unpopular dictators and monarchs when you come across them to help out the locals and play world police.