I'm not sure about the visa part, but about the time:
By the cheapest train I seem to recall Narita airport is about two hours from Tokyo. Also you might well have to be back at the airport at least one hour before boarding, and quite possibly more. And the train system in Japan is notoriously complex. It will be very easy to get a bit lost and miss your flight in Tokyo (I've done it once) and it will be very expensive to just take a taxi the whole way to the airport from Tokyo. You might consider renting a car, but you'll have to make sure you've prepared in advance because even with English on all road signs, Japanese place names etc can be a bit bewildering when you're not used to it.
If you just want to dip into Japanese culture though, don't worry because Narita has quite a bit to offer for an airport city. It's very compact, hard to get lost in, and close to the airport so even if you have to take a taxi it won't be too expensive. It's a fairly typical little Japanese city with lots of small shops, food, parks, a calligraphy museum, and a temple that is at least a thousand years old.
My first two times in Japan I had only 24 hours in Narita between flights and I enjoyed it very much.