Tokyo is not cheap.
For the most part, it is ridiculously expensive.
Most budget travelers interested in Asia opt for cheaper destinations like Thailand or Taiwan.
However, it is possible, with a little planning, to travel cheaply in an around Tokyo. While you will not find $3 guesthouses and twenty-five cent lunches, the Tokyo experience is not as out of reach as many people might think.
There are dormitory style hostels similar to your typical youth hostel or YMCA. People of any age can stay in such places. There will be little privacy, but they are well run and reasonably quiet.

A uniquely Japanese invention is the capsule hotel. These rooms are extremely Spartan, sometimes offering a chair or sofa in lieu of a bed. Shower and toilet facilities are usually found "down the hall." These are the cheapest accommodations, but, according to those who have used them, neither very comfortable nor quiet.

There is the occasional cheap guesthouse in Tokyo. One dedicated to budget travelers is New Koyo. Beware of staying somewhere you have not researched first. Cheap is often synonymous with sleazy in Tokyo.

A web site titled Planet Tokyo offers lots of useful info for those interested in touring the world's biggest city.