Are there any places I can take an FAA written test in Japan?
According to this list, there are three Airman Knowledge Testing Centers outside the 50 states (listed as "Out of US" on that document.)  All of them are in US outlying territories, and none of them are in Japan.  

You could conceivably arrange do the written test at the center listed in Guam (Barrigada).  There are regular airline flights to Guam from Tokyo, Osaka, Fukuoka, and Nagoya, and the 4-hour flight to Guam is better than the 8-hour flight to Hawaii or the 9+ hour flight to the continental US.