For two people, cheapest place I know of is Ryokan Tokaisou in Asakusa.
http://www.japanhotel.net/2005/room.asp?HotelID=564) AND it has an ensuite bathroom ^ .
For stylish and atmospheric rooms check out Ryokan Andon (you can easily walk to Asakusa from here in about 15 minutes). It's one of the nicest places in Tokyo for your budget, and there's no curfew:
http://www.andon.co.jp/home/price/index.html - scroll down for some photos (press 'design' to see the building)
- thanks jib71!
As well as
abmj-jr's excellent suggestion of japanhotel.net, you might like to try
http://travel.rakuten.co.jp/en/
Additional areas to try for cheap lodging are Nippori and Nishi Nippori (which service the gentile Yanaka neighbourhood - a great place to mooch around in early in the morning) and Uguisudani. (you'll find them using the 'Ueno, Asakusa, Ryogoku' option if you select 'Central Tokyo')
There's a whole thread on the 'budget travel' forum - just ignore the false start as the OP was enquiring about rooms in the 20,000yen + bracket. There are lots of 'real' budget options after the first few posts.
http://www.flyertalk.com/forum/showthread.php?t=622407
I have Australian friends who stayed at Kimi Ryokan and loved it. But I think people's enjoyment of this place depends heavily on how well you get on with the other people staying there. If you like the communal ambience of Backpacker Hostels you'll probably like it. If you want a more private 'hotel' experience leave it alone.
Having said that, places like Kimi Ryokan, Hotel Juyoh, Hotel New Koyo and the Sakura Ryokan (near Iriya) offer certain benefits to someone unfamiliar with Japan. I'm guessing Ryokan Andon will have some very interesting people staying there too. Dialogue with guests should give you lots of ideas and inspiration on things to do, and tips on cheap places to eat and ways to get around. The information is usually more pertinent than that given by a concierge (for instance, I would never have known about the 'Stamp Rally' opportunites if I hadn't spoken to a couple of German Psychiatrists at the New Koyo on my first trip to Tokyo.)