"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 US Embassy has two hard courts in its housing compound. Nice setting, sort of at the top of a hill with a modest view of the neighborhood. You need to know someone (or the spouse of someone) employed at the US Embassy who will invite you to play with them there. Or, Krissman International teaches tennis to both employees and non-employees at these courts; if you take lessons you may have a shot at making friends with embassy players...
Location is between Roppongi and Tameike on Roppongi Dori, on the opposite side from the Ark Hills complex.