Rotax is a good bet in Japan. Last weekend alone for example there were 3 events around the country that were part of the Max Challenge.
A Rotax-type engine is pobably good to start with and fast enough that you won't get bored. Remember that karts vs cars and motorcyles is a bit different. I've raced all three - don't expect to be fast immediately in a kart

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There also seems to be a category in some parts of the country at least for the X30 engine, grids ranging from 5 to 15 at least at the Motegi circuit.
Yamaha should be popular with big grids. Might be a bit slower than what you're looking for though.
There's also the All-Japan Kart Championship which includes a KF2 category - which is faster than Rotax. Grids don't seem to be that big from the results i've checked though.
If you need the different categories explained a bit more let me know.
One of the main Japanese websites is
http://www.eikoms.com. If you don't know Japanese it might be a bit difficult because it's almost completely in Japanese but using some translation services is enough to get around the site and see what's going on

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