I'm sure several Old World cities like Damascus will be mentioned, but in the Americas, the actual structure of Taos Pueblo in northern New Mexico dates back (probably) to the 13th-14th centuries AD. Obviously, people have had to maintain it to continue living in it and there have been additions, but the location and much of the structure are based minimally on the existing structures in 1590s and probably extends back to the 13th-14th centuries.

Acoma Pueblo in western New Mexico has a similarly long occupation, but I don't believe any of the structures themselves date back that far - just the occupation of that location.

Im sure it isn't the oldest, but a really neat and well preserved section of medieval city is in Sighisoara, Romania. It is a UNESCO world heritage site. It is still inhabited by nearly 30,000 people and has been continuously since at least the 14th century. The oldest settlement in that location was in the 12th century.