Places to play: Russian Embassy
"It's who you know..." goes the adage. There are a number of tennis courts in central Tokyo for which you need a connection in order to use the courts: at embassies, international schools, companies and others.
The Russian Embassy has a hard court in its Embassy/housing compound next to the Tokyo American Club. There are some tennis-playing American Club members (me included) who look with envy on that court, from the American Club bowling alley windows, since the American Club doesn't have its own courts. TAC's tennis group (again, me included) has played with Russian Embassy friends at this court, and it's a pleasant place. The Russians are pleasant too, and they've got some very good tennis players.