Important geography notes that I haven't seen covered in regards to this series - in English, anyway -and I think are important for understanding.

Yozakura Quartet is set in Tokyo. Sakura-shinmachi is a neighborhood in Tokyo. And yes, it is Sakura-shinmachi, not Sakurashin Town. I got the romanization off of a train map of Tokyo, and it's also romanized on the train stop in Vol 5 of the manga. Sakura-shinmachi really exists. [url= http://en.wikipedia.org/wiki/Sakura-shinmachi] It really is two train stops west of Sangenjaya, or Sangenjaya no nishi no nishi
Hime is 町長 chou-chou, town/neighborhood mayor. Yuuhi would be 区長 ku-chou, or Ward Mayor. Wards in Tokyo are like boroughs in New York, except there are 23 instead of 5. Yuuhi would be mayor of Setagaya Ward. Setagaya Ward is right next to Shibuya Ward, and Sakurashin-machi is four stops from Shibuya Station. However, the express train doesn't stop there, so you better get the local.
Tokyo itself is referred to as 都 (to), or metropolis. The "mayor" of Tokyo is referred to as a Governor in English. The 23 wards each function as their own cities, and in fact they even call themselves cities in English. Shibuya-ku is City of Shibuya, Setagaya-ku is City of Setagaya, etc.
So, in short:
Tokyo 都 (to) = giant megapolis. 都知事 to-chiji = Governor of giant megapolis.
Setagaya 区 (ku) = borough or city. 区長　ku-chou = City or borough mayor.
Sakura-shinmachi 町 (chou)＝ town or neighborhood. 町長　chou-chou = town or neighborhood mayor.