Obviously it's hard to quantify "how much" they knew on the subject, but Rome was, in my experience, the most prominent historical empire, and the average educated person would be expected to know quite a bit about it's history. For example, I recently translated a fairly popular 12th century encyclopedia, the Imago Mundi of Honorius Augustodunensis (fun fact: Columbus got much of his geographical information from Honorius). The third book of the encyclopedia concerns itself with world history and within Rome is by far the most prominent historical empire. This is true both in the amount of space dedicated to Rome and the "flavor" of how history is presented in general, almost certainly as a result of Honorius borrowing from people who had borrowed from earlier Roman historians. The intellectual development of Christianity was deeply entangled with the Roman world, and since knowledge of Greek had largely died out in the west, Roman sources were the basis for education.

There's also a sense among medieval thinkers that the Roman Empire hadn't really ended, so much as suffered a brief interruption after which its legacy was continued by the twin institutions of the Holy Roman Empire and the Catholic Church. Honorius is illustrative here as well, his list of Roman consuls and emperors ends, not with the fall of Romulus Augustulus, but in the present day (c. 1130). Roman history was thus understood the history of the empire in which many medieval authors were currently living. Just as everyone in America is familiar with George Washington, it's reasonable to assume that everyone (educated) in the empire was aware of who Augustus and Romulus were.

Just to follow up on what has already been said here: the Byzantine Empire was still the Roman empire in many's eyes as it was the eastern half of the Roman empire. They still called themselves Romans. However, Justinian was likely the least Roman Emperor to speak Latin as a primary language, as an Empire centered around the Aegean naturally became Greek in almost every way. So, technically it was still the Roman empire.