Edit: BubaMorg summed up exactly what I was going to say, and even more!
Remember, you CAN NOT get a 'foreign residents' ID Card on a tourist visa. That is your alien registration card, and it is needed for almost anything dealing with life living long term in Japan, IE bank accounts, housing, credit cards etc... You can not even get a cell phone with out one (the pay as you go ones are hard to find, i never did find one)
If you were Canadian or a citizien of a common wealth country (Australia, New Zeland, UK) then you can stay and work for a year no probs on what is called the WHV Working Holiday Visa. However Americans are exempt and will be denied a WHV so you are SOL on that one.
If you want anymore info, then feel free to pvt message
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I have been in Tokyo for over a year now, and I had to go through probably the toughest things, visa and apartment hunting as a foreigner here.
I skimmed the thread briefly, some good advice, some incorrect advice.
It's been the best year of my life, so I encourage everyone to take the chance and come out for awhile and see what goes on outside your town, county, state, border