Gaijin House´s:
Home
the one i stayed (quite nice!)
Oakhouse | Guest House Tokyo, guesthouse, room rent in Japan
and a site with appartments so that you could see into the prices. They might have something reasonable but im not sure:
Tokyo Apartments -Reasonable Apartment Accommodation Guesthouse
As you are a native speaker i think your best chance would be to find work as a teacher or give private lessons. I did the WHV year but didn´t teach. I got around somehow but i brought a bit more money... Be sure to check out
Japan Association for Working Holiday Makers once your in Tokyo, maybe they have a job. Their monthly party is a good way to meet other WHV people.