When you change your address within Japan, you are required to update the address on your driver's license as well as your Alien Registration Card. City Hall doesn't handle this request. Rather, all driver's license procedures (besides the actual issuance of a license) are handled by your local police station.
Before you go out, though, make sure you've already updated the address on your Alien Registration Card at your city office. The Alien Registration Card will be your proof of identification when the police copy down your new address to your driver's license.
You'll need to find out which station's jurisdiction you're in, then visit that station during it's open hours. The word for police station in Japanese is 警察署（けいさつしょ）, and you can put that term to use by searching for your city or town name + 警察署 in regular Google search or on Google Maps.
Police stations are usually open on a schedule similar to the immigration bureau and other public offices: That is, they start opening around 10:00 and briefly close for lunch between about 12:00 and 13:00. My local station stays open as late as 17:30, but they stop accepting applications for procedures around 17:00. It would be wise to arrive as early as possible or to call ahead to check your station's open hours.
Driver's license inquiries will be handled at a specific desk. Usually it will be clearly marked with a sign that says something like 運転免許各種手続き（うんてんめんきょ かくしゅ てつづき）. If you aren't sure, find a friendly-looking receptionist and pull out your driver's license. Someone should direct you to the right location. Show them your license and your Alien Registration Card, and explain that the addresses are different. (Or, alternatively, show them your license and your Alien Registration Card, and frantically and dramatically point at the cards' differing information. They'll get the message.)
By the way, if you are lucky (?) enough to live in a city with a regional licensing bureau, the bureau will handle all license-related procedures for local residents. Because of that, the police stations in these cities do not handle license-related procedures.