Places to play:  Shinjuku Station / My City Building rooftop

This is primarily a tennis school.  You can rent the court when the school sessions are not taking place.

The artificial-lawn court is a singles court with not much space around it - sharply angled shots would be hard to get to, as would be deep high bouncing shots way behind the baseline.  The enclosing net also precludes high lobs.  

The rental court rates per hour are 4000 yen weekdays from 1:35 PM, 6000 yen weekdays from 4:15 PM, and 6000 yen Sundays from 6:30 PM.  Junior lessons are 4:40 PM on Wednesdays.

The school has beginner and intermediate classes. You sign up for a unit with 8 classes generally done within one month. One unit is 10,000 yen for weekdays and 11,000 yen for weekends and nights.  There is a first-time registration fee of 5,000 yen.  Private lessons can be arranged.  Coaches might be able to give simple instructions in English.

The My City building is the building above the east exit of Shinjuku station.  Take the elevators at the south end (Yoyogi end, farthest from the east exit plaza area) to the roof.  Phone is 3354 5133.  Calling in Japanese is probably best.