Road names is not my thing but I note there was actually a discussion about this some years ago, which you can find here: https://www.reddit.com/r/AskHistorians/comments/61ll60/did_the_romans_name_all_their_roads/
At Rome, and in other cities we can find clear evidence of names given to roads, whether from inscriptional evidence or in other written documents that use the names in passing. The best evidence always comes from Rome, and there the case is pretty clear: yes, roads had names.
Regarding sending a letter, however, you wouldn't address it the way we would today. Letter sending was a very personal business - you won't put it 'in the post', because there wasn't a formal postal system (with exceptions I'll outline below). So if you wanted to send a a letter, like in your example, from Italy to Asia Minor, you'd have to either find someone who was already going there, or send someone specifically. Even by the powerful, both options would be employed - so something important my be entrusted to a slave or friend and sent out immediately, but something less so might be written and then held onto until the person who wrote it found someone who was going that way. Wealthy individuals often had a staff of designated letter carriers who could do this work for them (tabellarii).
There's some really nice letters preserved from a military camp in Britain called the Vindolanda Tablets. My favourite of these, and much the best known, is a letter from the wife of the commander of the Vindolanda fort to the wife of the commander of another nearby fort inviting the latter to the former's birthday party. On the outside of the letter, it is simply addressed 'To Sulpicia Lepidina, wife of Cerialis, from Severa'. So Severa would have given that letter to someone at Vindolanda who was going to this other fort, and then would have tasked them with tracking Lepidina down (in this case, extremely easy because she was that fort's first lady). In this instance, Severa was probably also taking advantage of military communication systems that would be in place already to allow communication between forts. But back to your Capua-Ephesus example - it would be a case of giving the letter to someone you trusted and saying "get to Ephesus and give this to X".
That said, there WAS a formal imperial postal system that had emerged under the Republic (Empire before emperors). This was a system of relay stations that allowed messengers to move across the Empire at speeds up to about 250 miles per day, or allowed important travellers to travel in greater comfort. But you had to be on official business or have imperial say so to use this system, you couldn't just pop a letter to a friend into the imperial post!