Usage is through the Tokyo Metro lottery system, which is described in a separate listing.
The address is Yashio 4-1-19, Shinagawa-ku, and admin office phone is 3790 2378. The location is just to the east (bay-ward) of the Oi horserace track.
Getting there by car (your own or taxi) is easiest/fastest. A fairly straightforward path is to get on the Dai-Ichi Keihin route (also know as route 15) that passes in front of Shingawa station, follow this south (towards Yokohama) about 3.5 kilometers from Shinagawa station, then turn left (bay-ward) at the Minami-Oi-Ichi intersection. Follow this straight about a kilometer, passing under an expressway and over a short bridge. At the first large intersection past the bridge, turn right and you are in the park. The courts are in two separate clusters; the admin office is at the cluster closer to the baseball fields. (By the way, other intersections before and after the Minami-Oi-Ichi intersection will also get you onto the island.)
The closest train is the monorail going to Haneda airport; get on this at Hamamatsu-cho station and get off at Oi Keibajo-mae (the horserace track) and walk 5 to 10 minutes across the bridge onto the manmade island.