You could try the Tokyo International Hostel:
I've been here and it was clean with english speaking staff, however be careful as most hostels like this have an 11pm curfew and you won't be able to get back in until 6am. I've also booked through wotif.com in Tokyo for standby rates on hotels and found some good deals. over 10 years ago
I have stayed at a few spots in Tokyo.
A very cheap Ryokan in Asakusa is Taito Ryokan.
http://www.libertyhouse.gr.jp/ Its pretty low end but the area is good and at 3000Y you wont get much cheaper. It easy to get to from the subway. (There are other ryokans in Asakusa, do a search) Also, ask for the public bathhouse (sento) its quite an experience but a bit hard to find.
Something (a lot) better is Hotel Kitcho in Nihonbashi which is an older part of Tokyo. Its a bit of a trek coming in fron Narita but once you are there it is on 2 main subway lines. It also has a japanese bathroom in the room and also an electronic toilet. (washlet) Cost was about $100au. I liked this one!
Eating in Tokyo is great. As well as the usual expensive places there are lots of eating bars (for businessmen) where you buy a ticket from the vending machine press the button near the picture you like. You take the ticket to the cook and wait until your number is called. (in Japanese)
Good luck about 10 years ago
Sakura Hostel is located in Asakusa, at the center of shitamachi (historical / nostalgic area of Tokyo). Not to mention the convenience to explore historical spots like Senso-ji temple or Kaminari-mon on foot, subway Ginza line（銀座線） and Asakusa line（浅草線）and newly opened Tsukuba Express line（つくばエキスプレス） are available from Asakusa station, and most tourist attractions in Tokyo are easy-accessible.
Web site: http://www.sakura-hostel.co.jp
Wide variety of room types - dormitory / twin, 4 ~8 bed private rooms
- 2,940 yen per bed. Male, female, and mixed rooms.
● Private rooms
- Twin room: 8,295 yen per room
- 4 bed private: 12,600 yen per room
- 6 bed private: 17,640 yen per room
- 8 bed private: 23,520 yen per room about 10 years ago
I highly recommend Tokyo bnb ( http://tokyobnb.com ). It's a newly open 3bedroom house in the perfect Tokyo downtown location of Roppongi. Much much more convenient than asakusa. The rooms are new, clean and uniquely decorated with Japanese arts. Because of small number of rooms every guest can relax and enjoy their stays. Their rates are also highly competitive. The bnb is run by a friendly and helpful Thai lady who is ready to help you with any question you have. about 8 years ago
I recommend Washington Hotel and Remm Hotel at Akihabara. They are cheap, right next to the station (literally next door), and very clean. There is also a 24hour Dennis right at the 2nd floor of Washington Hotel, very convenient when you are hungry at night due to jetlag. There are also plenty of restaurants nearby =)
Remm: http://www.hankyu-hotel.com/cgi-bin2/cms2/index_en.cgi?hid=51remmakihabara about 8 years ago