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 indoor courts see the info under Jingu Gaien Indoor Courts.
For the outdoor courts: These are hard courts, set amongst a lot of greenery. They are next to several baseball diamonds, so after a dry spell and with some wind, the courts can be pretty dusty. Basic fare, but it's tennis. Main sign-up day is the 16th of each month for time slots in the following calendar month; for popular slots, sequential sign-up numbers are handed out from 4 PM on the 15th on a first-come first-served basis. Few of us can get to that sign-up process, but take heart -- there are generally some slots left. Andy Noble and Jim Hilgendorf found time slots to be abundant if they got there soon after the 16th of the month.
For weekend play the pre-10 AM slots and the post-5 PM slots seem to be frequently open on one court or another. These slots get gradually taken too, the more distant you are from the 16th.
Cost is 3000 yen for 1 hour up to 9 AM, then 5000 yen per hour. No visitors fees ! Wear any color of clothes you want ! You must pay at the time you reserve; you'll be given a receipt showing court time and payment No refunds except for rain-outs. Of necessity, you'll have a chance to practice Japanese with the staff at the reservations office.
You can check open times at 3403-0923; they will hold a reservation for one day only, within which time you have to get there and pay. Changing rooms available near the reservations office.
Directions to both locations: 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. Continue on the tree-lined boulevard until it ends at a curved road. Just across the curved road is a fountain; beyond that are several baseball diamonds, and beyond those in the distance is a square stone building (the Meiji museum). Take the curved road to your left, staying on the inside of the curve of the road. About 200 meters from the fountain is a cluster of three 2-story buildings. The reservation office is in the middle building on its baseball diamond side. The courts themselves are strung out end-to-end on the opposite side (east side) of the baseball diamonds from the reservations office. There is a fence blocking entry from anywhere except from the direction of the reservations office, but you have to go to that office first anyway before you play, to show your receipt.
Coming by foot, bicycle or taxi is recommended since parking is tough in this area on weekends.