What is the cheapest way to stay and travel in Tokyo, Japan?
Go to any hardware store and buy a "blue sheet" - a plastic ground-cloth, some pegs, well buy a few sheets, some cord or string, maybe some buckets, whatever.
Head for the big green park around Meiji Jingu in Shibuya, follow your nose to the tent village and set up your new Tokyo residence. Join the homeless - meals can easily be scrounged within a quarter of a mile if you don't mind dumpster diving.
You could also spend a few first days and nights pub-crawling and finding a nice person willing to put you up as a "housemate" and "language companion." Swap room and board for your linguistic and cultural skills. A win-win proposition.
Got some dinares (money) to spend?
International guest houses are really pretty cheap. But so are capsule hotels. Thinking of a longer stay - look for a share-house arrangement. Make friends on the cheap. Hey stranger.
Hitchhike. It is less impossible than what you may have heard, and leads to new friendships and often a meal or roof over your head.
Take the long slow trains, maybe at night. Hard seats but plenty of time to soak up the countryside. Get off where you want, find an information booth and ask for any youth hostels in the area - they are forgotten relics but often still there.
Get a bicycle, cheap, or a second-hand motorcycle, pretty cheap. Head for the horizon and the open road, Easy Rider style.
Cheapest legal way, depending on your age, would be to participate in a home stay opportunity, or stay with friends.
You could also maybe do a home swap with someone from Tokyo what wants to stay in your home in exchange. There are sites for that.
Other options could be to stay in a capsule hotel or cheap minshuku or ryokan.
To get around, you can just walk or ride a bike, or a scooter, you can buy them secondhand.
You could price up a pass for the subway if you are say staying for a month or longer, or a JR rail pass overseas, but they aren't necessarily the cheapest option.
Car ownership is expensive due to registration and the need for a parking space plus fuel is expensive there too.