Correct me if I'm wrong, but I'm 95% certain that the process of applying to JET, being reviewed, being accepted, and getting on the plane is much longer than 3 months. In fact, I know that JET teachers leave in the summer time, and in April if they apply for early departure and are accepted for that. So. It's september. April 2013 is six months /180 days away.

If you get on the phone now and start making appointments your cat will only be quarantined for a few days (if you happen to leave April 1st). If you're like most and don't accept early departure you have plenty of time to get the import conditions dealt with. If you don't get accepted... well, congrats, your kitty is ready for international travel.