Here in Japan right now, I seriously recommend travelling with somebody else, that makes it way easier to go eat in decent places. Not that they'd kick you out, but I always feel like an asshole going to an actual restaurant alone.
For places to live, you can get AirBnB pretty decently depending on location, and are usually the preferred option. They are bit pricier than hostels, for extended stays I like to have door behind which to keep my stuff locked in.
Capsule hotels are good(if you stay in decent ones), but only stay in those for one or two nights. There's practically no room for luggage, and they really aren't cheap enough to consider over any other option.
Manga Cafes or DVD places are also an option, but only do that if you want an experience, and if you do, prepare something to block your eyes with. They never turn off the lights. And anyway, overnight stay in one will be more expensive than a hostel.
If you want to live cheaply, go to supermarket and buy stuff to cook. But Japan is relatively cheap, if you budget 500(around 5 dollars) yen a day for breakfast from Convience Store, it should be plenty.
Convience Stores in general are very good, and you can grab most things necessary from one. Also, rather than trying to find a place in which to exchance money in, just get cash from machines in 7/11s. Exchance rates around Tokyo pretty much universally are not great, compared to automatic machines at airport you'll pretty much only find something that offers better rate by 0.4 yen.
If you are going to be travelling a lot within the city, either get ready to walk a lot(not that good idea in winter, the weather here is very annoying dry-cold) or budget around 1000 yen a day for trains and metros. It will be more than what you'll need on most days, but it's always better to overbudget.