Virtually all of our literary sources on ancient Rome were transmitted through the Christian monastic system. The oldest literary manuscript we have is the Vatican Vergil, which dates from the early fifth century CE. There are a small handful of manuscripts from about that time period, but I think they are all of Vergil and the Bible.

If you want texts that are older than that you are in the realm of what I consider archaeology, or at least epigraphy. In Latin, there are a handful of extended literary compositions preserved in inscriptions--the most famous is the Res Gestae Divi Augustus, an extended (what I would call) pseudo-autobiography of Augustus. There are a few things like this (Claudius' Lyon tablets being another well known example) and there are also examples of poetry on tombs and the like. But outside of these few lucky finds, the vast majority of the corpus of surviving non-transmitted Latin are dedicatory inscriptions, tomb epitaphs, etc. There are also a few examples of administrative records, such as the Vindolanda Tablets.

With Greek we have a few more sources, and not only because the corpus of transmitted Greek literature dwarfs that of Latin. For one, Greek was a common language and culture of Egypt, and Egypt's hot and dry environment preserves manuscripts, giving us some rather extraordinary "new" texts (such as Menander and the Oxyrhynchus historian). Added to that, there are many places in the Greek speaking parts of the empire with a very vigorous epigraphic habit, meaning we have more examples of things like civic history from the Greek world. But even with Greek, these finds pale in comparison with the amount of transmitted literature.