I hear that it's possible to get alien registration card from a city hall without any kind of visa to reside in Japan... so I suppose it might be possible to go to the city hall and get an alien registration card which you can use to buy a phone...

(I recall that I had to do this when I first came to Japan and stayed more than 90 days on a tourist visa. I was allowed to extend the visa beyond 90 days - but I had to go through the alien registration business with the local city hall).

Rather than going through all of that hassle - The easiest solution is probably to ask an acquaintance with the necessary documents to sign up for a prepaid phone. If you're spending any length of time here, it should be possible to find someone who is willing to do that for you.