What about Youth Hostels?

Youth hostels are among the most inexpensive types of accommodation in Japan. One night typically costs between 2500 and 3500 yen per person and usually 1000 yen more for non-members. Memberships can be purchased at the hostel.

There are over 300 youth hostels spread out across Japan. They do not differ much from European or American youth hostels, except that breakfasts and bathrooms may be in the Japanese style.

http://www.jyh.or.jp/english/index.html
http://www.japan-guide.com/e/e2030.html

Warm regards,
Buffy

OMG!! I am surprised you can make heads or tails of all this with all the conflicting and grey information over this matter. I hope that you find the resources that you need.
Dave my friend from my New Beginnings church mentioned the New Hope church in Tokyo that I had talked to you before. Perhaps they might have information and connections helpful for you.

New Hope Tokyo Office
Information: #308　Tazawa Building 2F
Sinjyuku-ku wasedatsurumaki cho
Tokyo Japan
TEL:03-3209-2170
Email: http://newhope.jp/en/about-newhope-tokyo-en/contact-us-en
Website: http://newhope.jp/index.php

I talked with Dave tonight and he said he is going to talk to his sister in the next few days whose Japanese blog I showed you and who had lived in Japan for a couple of years. He said he will let me know if his sister has any suggestions for you.

I pray you are able to secure safe and affordable housing in a timely fashion.

Warm regards,
Buffy