[Nyororin]

You asked this at a very good time. I don`t have any dogs, so normally I wouldn`t know this... But yesterday the fliers came around for the dog vaccinations/fees. So I just happen to have the paper on hand.

The registration itself appears to be about 5000 yen, with yearly vaccinations at about 3000 yen. When you move, you have to register again and pay the new fee in your new town.

Stray cats aren`t too rare, but stray dogs are. You certainly won`t be able to go out and just find a dog on the street. If you were interested in mix-breed puppies though, it`s quite common for people to give them away. There are some sites online that show lists and give contact information so that you can take one in.
I can`t say I recommend bringing a dog over here. I`m not 100% certain for dogs, but for cats it is 20 days of quarantine at 8000yen/day plus food costs. We decided not to bring my cat from the US because of the length and price. I doubt it`s all that much different for dogs - maybe even longer and more expensive.

Most apartments in Japan do not allow pets. And often those who do will allow rabbits and cats, but no dogs (as the barking may disturb neighbors). You may have a lot of trouble finding a place, and it will probably be more expensive than a normal apartment.