how long to stay?

To really understand a foreign culture, you may have to stay in that country for a year or more. Many people do this as exchange students. For shorter stays, check with the Japanese Consulate for a list of Japanese families that will take you in for a week or more. Some of them provide free lodging and meals, in exchange for doing chores at their house. That is, assuming you already know some conversational Japanese. To go to a foreign country not knowing their language can be very hard.

20 months,

Let's just say, the longer you stay the more things you'll discover.
I was in Japan for 3 months (went to a language school) and I was still finding new things even up until I left.

So with that, I guess your stay depends on how much you can afford.

For accommodation I suggest staying in a hostel. They are very cheap, you can probably find one for $20/night. If you avoid eating out for every meal, you'll save money in that department too. And getting a rail pass will save you TONS on transportation.