The lines are a combination of actual, named roads (like the Via Appia or Via Militaris) as well as roads that do not have a known historic name (in which case I creatively invented some names). Skip to the “Creative liberties taken” section for specifics.
How long would it actually take to travel this network? That depends a lot on what method of transport you are using, which depends on how much money you have. Another big factor is the season – each time of year poses its own challenges. In the summer, it would take you about two months to walk on foot from Rome to Byzantium. If you had a horse, it would only take you a month.
However, no sane Roman would use only roads where sea travel is available.