Beyond an undergrad degree, the qualifications vary from job to job. Your safest bet would be to get a TESL certificate (online or through your uni) before you start applying for jobs.

A lot of jobs in Japan do their hiring in March/April. If you're applying outside the "hiring bubble," you may have tons more success if you go to Japan and start looking on site. I've heard great things about the Kimi Information Center in Tokyo - you can look at job postings and get help with the basic survival stuff, like navigating the subway or scouting an apartment. Good luck!

you don't have to have to a degree in Japanese Language / Culture to teach in Japan, although that will certainly help, as long as you have a degree in some subject you are eligible. The TEFL or TESOL along with a degree will almost guarantee you a job over there (according to friends who have already taught in Japan)