I was reading Lest Darkness Fall which is about a 20th century man sent back to around 500ce to the city of Rome and he knows classical latin so he communicates initial by writing his requests down. This stuck me as odd, what are the chances whoever you meet in rome is literate.
As the scene is presented in the book, I would say Padway got pretty lucky in happening across someone who could read. With just a bit more detail added, however, he would have increased his odds of hitting upon a literate person dramatically.
In the early fifth century, so a bit before our time frame, an education in reading and writing classical Latin was standard not only for upper class men but also for upwardly mobile middling types. Late antique Rome, with its far-reaching empire, was quite bureaucratized. By 400 education emphasized rhetoric and logic rather than classic literature, perhaps, but literacy was still an important skill for making something of yourself in Roman administration.
Now, of course formal education to the extent of comprehending Cicero was out of reach of the vast majority of the Roman population. But Christianity had introduced its own value on reading (which survives into the Middle Ages in monasteries). And there is also the nature of living in a literate society: levels of literacy. A random Roman soldier might not understand Tertullian, but he might very well recognize the format and a few key terms on his travel permit.
By 600, however, the disintegration of the edges of the Roman Empire in the west combined with a militarization of the levels of government to make imperial bureacracy much less important. (Although in Egypt, it seems to have held together better for longer.) Governors still relied on letters for diplomatic communication, and some aristocrats still saw literacy as an important value (especially for religious purposes). But in general, by that point reading had lost its social value and much of the immediate need.
Naturally, de Camp sets his story between the two points when we have a somewhat decent idea of the situation! Still, it's safe to posit that in 535, a random slave picked off the street in Rome certainly could not read free Latin prose. A woman? We know of some excellent female Latinists of late antiquity, but many fewer women learned to read than men. But had Padway selected a man in clothing that clearly marked him as an aristocrat? Sure, in 535 there is a decent enough chance that such a man would comprehend Latin prose. So let's assume that Padway simply forgot to clue us in on the detail of the bread-carrying man's clothing, what with all the mental effort of trying to think in Latin.
Did you mean BC? I think Rome had "fallen" by 500 CE.