Bury, cremate, ship body--all three are possible. Generally speaking for the Japanese, the body is cremated (very high temperature that the bones left are very clean and generally white and most of the fragile bones turn to ash) and laid to rest in a grave (bought either by the family or oneself in advance). This is common practice now regardless of being Buddhist or Christian. (I have no info regarding Muslims.)
Foreigners laid to rest in the old days were buried in the Western/Christian style (uncremated), but with the lack of space, many Christians opt for cremation in a small plot. But, there may be people willing to pay for a grave to bury someone uncreated. This depends on how much money you are willing to pay for the grave.
The Foreigners Cemetery in Yokohama mentioned in the thread no longer has space for new graves. The graves there are mostly of people who passed away in the Meiji/Taisho periods. I know a family who insisted on having their father buried there (in the 1970s), but it was done at tremendous cost and by pulling a lot of strings.
I have read of foreign nationals who have died in accidents in Japan having their uncremated bodies shipped to their home countries, on request by their families. So, I believe it depends on what the surviving families want.