Japanese law only says those who who do not have a Japanese address, hotels are to record passport information.
If you have a Japanese address then you do not have to show your passport.
The problem is some hotels don't seem to understand the law very well and request all foreigners show a passport. By stating you are a resident of Japan and writing in your Japanese address at check in should be enough.
Here's the law in Japanese if you wish to print out a copy with you:
www.mhlw.go.jp/topics/2005/03/tp0317-1.html
(If you can't read Japanese, the main part first in the color red under part 2, which states only foreigners who DO NOT have a Japanese address are to have their passport information recorded)
If they ask and if you want to, you can show them your Alien Reregistration Card or Resident Card (which you are supposed to carry with you in case you don't have your passport on you), but that's up to your discretion (showing it to the hotel that is). The hotel cannot legally compel you to show your Alien Registration Card/Resident Card if you already have a Japanese address.