You really should play Shenmue II. Well worth your time.
Shenmue Gai is the official title of the game in Japan, and Shenmue City is a wrong English translation. Suzuki said in an interview that there are plans to move the game to the western markets, and its tentative title was Shenmue World. These plans are not concrete and have not been elaborated on at all.
Shenmue Gai is basically a text adventure game developed by Yu Suzuki's new studio YS Net out now in Japan on mobile platforms, and will soon (no release date yet) be coming to Yahoo!'s Mobage Town service. SunSoft is publishing the game, and SEGA is I guess supervising it, and DeNA, another company, I think is co publishing? Not sure how what their job is in the project.
Gai starts off in the world of Shenmue I, features quests from the original Shenmue I, as well as stories that were cut from the Dreamcast original, for whatever reason. It is also regularly updated with limited events and capsule toys, which is what makes money for the game. In interviews, Yu - san has said that if the game is successful, we will see a continuation into the world of Shenmue II. I'm guessing we won't see that until after the game is released on Mobage Town.