You get the card at the airport, but it's useless without the stamp. Can confirm that the dependency graph stated by the parent is a (the?) correct way to navigate the system. The only thing I'd modify is that you can save money by bringing your own phone, and getting a phone+data SIM card from Bic, which you can set up on a recurring plan with a foreign credit card.
Yeah, it's been so long that I actually forgot the proper (I think) way to go through it, and that it was actualy easier for me because I had contacts/friends willing to help.
The problem with the phone+data sim thing is that you need an actual callable phone number, a lot of the time. Also, you can tell what numbers are used for in Japan by how they start: https://en.wikipedia.org/wiki/Telephone_numbers_in_Japan#Non... , and some places will be sticklers about it.
One sad/funny thing is that I actually needed a phone number to get a place to live as well, but actually my agent just used their own temporarily (literally right in front of the leasing agent, and they didn't care). I get the feeling that if you get a place from abroad they'll overlook that/work that out somehow, even if you don't have an agent that was as understanding/nice/flexible as mine.
Oh, if you have a phone+SIM card from Bic, you get a number. You need a non-tourist visa for that one, though...which means you need the stamped gaijin card.
I never had issues with anyone being snobby about the number, but I was in a bigger city, so maybe I had it easy.
(My sad/funny story was that in order to cancel the recurring phone payment, I needed to receive a text message to the phone number I was cancelling. But I had left the country by that point, and the SIM didn't work for international SMS. So I was screwed. I just told my credit card company to reject the recurring billing, and thus I'm probably blackballed from all future SIM-card purchasing in Japan.)
