Be flexible and plan ahead!
Rule number one: plan ahead. It's possible to do Tokyo affordably, but cheap accommodation gets booked up fast. If you want a super-cheap place, be prepared for tiny spaces and locations that are less than ideal.
If you're REALLY strapped for cash, you'll either need to stay overnight in a manga cafe (around 1500 yen for a small cubicle with a mattress in a large room full of other patrons), or camp covertly in one of the larger parks (not really an option in winter). Couchsurfing.com is another option for free (for members) roomshares.
If you want to spend around 3000 yen per night or less, Airbnb.com has a few options in Tokyo--check them out for affordable roomshares in the 3000 yen per night range, and a smaller number of beds for 1500-2000 yen per night. As always, use services like Airbnb and couchsurfing at your own risk, though both have good reputations and means of vetting and verifying hosts and guests.
Sakura House has been around for a long time. They operate multiple hostels and guesthouses all over Tokyo. Rooms vary in price and quality (at guesthouses your housemates make a big difference), but rates are reasonable, rooms are easy to book, and staff are helpful. Dorm beds are around 3000 yen, single rooms around 7000. (www.sakura-hotel.co.jp)
Capsule hotels are more of an "I can say I did it" sleeping experience than a comfortable hotel option, and they're frequently not much cheaper than business hotels (around 6000 yen per night). Most only accept men. And if you're over 6 feet tall, you're not going to get much sleep.
For a quasi-capsule experience that's a lot more comfortable (and cheaper at 3500 yen per night), check out the Capsule Ryokan in Kyoto, a friendly place that also offers very nice hostel rooms with space-age showers (http://www.capsule-ryokan-kyoto.com/). The capsule beds are a lot more spacious, and the capsule room doesn't reek of alcohol or cigarette smoke.