I would recommend you getting another adapter and yes you can just plug in the new adapter on to your xbox.

Frequencies are various. In Japan there are 2 ratings depending on where you are, 110V either 50Hz or 60Hz. Most all equip supports both frequencies in Japan.

Step downs are indeed a good solution, but they eat up a lot of power and an adapter is indeed a better option. But since you have a home theater as well, maybe you should invest on a step down powerful enough to plug in your 360 and your home theater.

Make sure to check the power ratings though. 360 is around 200 Watts + Home theater power rating. Look for a step down with a higher rating then that and you should be ok. Also, don't forget to get an multiple socket extension from there, some step downs only have one socket and finding one here will be a headache.