Minimum Effort to Run a 1100W Hot Air Brush from America to Japan?
I think it will work as-is. It will provide about 30% less heating and similarly less air volume due to the slightly lower voltage, and frequency won't matter to the low voltage DC motor typically used (usually it's in series with the element with a bridge rectifier to provide the DC to the motor).

You could buy a standard 110:24V transformer, mount it in a box, and step up the voltage to get to full horsepower, but that's a lot of trouble, and might contravene some Japanese regulation or another if it's not done correctly and safely. It's 1100W so about 9-10A so you'd need a relatively expensive and heavy 240VA 120:24 50/60Hz transformer, and you'd need to wire it so that it adds to the mains voltage. It might cost more than just buying an ionizing hair dryer in Japan, which would  probably be pretty nice.

OP probably knows this, but Japan is part 50Hz and part 60Hz, pretty much everything West of where they are going is 60Hz, but their town (and Tokyo) are 50Hz.