If you need a quick shot of tennis, there are two locations in the Jingu Gaien area you may want to try. I'll not suggest that time slots are wildly available, but by central Tokyo standards you may be pleasantly surprised by what you can get a few days to a few weeks before the day you want to play. The locations are the Jingu Gaien Tennis Club's 8 indoor courts and the Jingu Gaien 5 outdoor public courts.
For the outdoor courts, see the info under Jingu Gaien Public Courts.
For the indoor courts: The court surface is carpet; the ball bounces low and fast. Being indoors eliminates rainy-season uncertainties. All-whites (white pants or skirt, and mostly white shirt or blouse) are required -- the first time there, I had to buy some white pants at the pro shop. Anyone can reserve these for hours between 3 PM and 11 PM on all days. Reserve up to a month in advance. Persons who are not members of the Meiji Jingu Tennis Club next door (i.e. just about all of us) must pay at the time of reservation. You might find out what time slots are available by calling 3401-7818 in Japanese. Court fees are 5,000 yen per hour. Visitor fee is 1,000 yen per visitor. So 4 visitors playing 2 hours pay a total of 14,000 yen. Changing rooms available.
Directions: let's use Aoyama Itchome intersection as a starting point (this is also a stop on the Ginza subway line). Go about 300 meters along Aoyama Doori towards Shibuya. You'll come to a large T intersection with a tree-lined boulevard leading off to your right. Turn right onto this boulevard. About 150 meters from the T intersection, on the left side, is a very large round-roofed building which has a parking lot on the first floor and the indoor courts on the second floor. Turn left at the short road just beyond this building. At the end of the road on the left (and the far end of the building, too), up some stairs, is the office for the indoor courts.
Coming by foot, bicycle or taxi is recommended since parking is tough in this area on weekends. The parking lot is the first floor of the indoor courts building, with typical Tokyo parking fees. The lot fills up quickly.