If you can get the working holiday Visa you should be ok to get smaller Jobs but mostly in teaching english and for a longer period. It will be hard to get any job just for 4 weeks!
Homestay´s in Japan are usually only available together with a language course, it will be hard to find one without a course. For a course + homestay (breakfast an dinner) you have to calculate around 160 000yen each. On top of that will be lunch, travel to and from school and your own spendings. However, i did have alot of fun in that time because you can meet many people.
Apartment: Renting an apartment in Tokyo for such a short period is not a good option. For Most appartments you will have to pay a high key money wich is not refundable! Your best chance would be to search for a big room in a Gaijin house.
Some links: School + Homestay
Geos Kudan Institute of Japanese Language & Cultur Tokio Buchung mit Online-Rabatt. Japanisch Sprachschule in Japan
General Prices and living cost
Prices and living cost in Japan
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 if you can rent it on a temporary visa:
Tokyo Apartments -Reasonable Apartment Accommodation Guesthouse