There are numerous good circuits in Japan, some are easy to get into (they are opened 5 days a week, and a race car can be rented on the place), some, as Suzuka or Fuji Speedway (the most popular japanese circuits), are open only in some moments since a lot of events are held there.
When you're in Japan, go to a book store and pick up a magazine about amateur racing. Look for the circuits advertised and give them a call.
(if you want to call Suzuka Circuit, the number is 0593-78-1111).
As for the car to rent, usually there's not a huge selection on the circuit. Just take what they give you... and cross your fingers it's an Evo or a Skyline (an r32 tuned maybe!!), but most likely it will be a Silvia, a Chaser, a JZX100 Mark II or a 180sx.
I personally love the RX8. It's excellent, the small 1.3l wankel engine and the settings make this car soooo easy to drive at high speeds!! Plus, it's incredibly sexy!!
.