As Montka said, you need
Airbrush
Compressor
Cleaning fluid (I recommend Tamiya's stuff. Easy to find here in Japan)
Some things for cleaning inside the airbrush (I confess I tend to use make up remover things, a bit like cotton buds, but pointy. The first ones were nicked from the wife, but now I buy em from 100yen shops.)
You'll also need
a hose to connect the compressor to the airbrush (this should come with the compressor)
thinner for your paint (I've had trouble using only water. Again, Tamiya do a good thinner)
Either an extractor cabinet (which you can build out of a kitchen extractor fan (you can buy em in K's Denki or similar for about ichiman en) and some thin wood (cost me about 2000yen in a hardware store.)
OR
A dust mask. Not one of the crap ones folk use for allergies. A proper hardware-store dust mask. Airbrushes send paint particles everywhere, and first time I used mine I ended up with blue snot. Not a problem in and of itself, but I cant help but think that blue lungs are not so good...
I used to airbrush indoors, with windows wide open and a fan, wearing a mask. When I came to move house I discovered that the entire surface of the room was covered in a thin, powdery layer of paint. That was a mega-cleanup job, let me tell you. Now I use my home-built extractor cabinet (set up simply so it sits on a desk with the fan end pointing out of the window.)
tbh, airbrushes are a bit of a bugger to get the hang of initially. They are undoubtedly great things, but be prepared for some wailing and gnashing of teeth before it all goes smoothly. It's worth it, though