The EEE PC either runs Windows XP or a Linux variant.
Both can display Japanese characters, but the EEE PC may not do so out of the box. I assume Linux would have a package available to allow for it (my Ubuntu installation displays them fine) and Windows XP can be easily configured to display them, although you may need access to a XP CD. Although the EEE doesn't have a CD drive, considering XP tends to ask for the file location if it can't find them, I see no reason you couldn't share it over the (wireless) network.
I have one. I've also used the Acer Aspire and HP Mini. If you can, I suggest you spend a little more and get the Acer or HP. The EEE is good, but the keyboard is just slightly to small to comfortably type on. The Acer is large enough, and has a larger screen. The HP Mini has a huge keyboard, yet the unit is small. For typing, the HP Mini is the best. Acer is better looking and easy to use than the EEE.
Anything that runs Windows/Linux can display the characters, you just have to install the font packages.
I have one of the eee pc's running Linux. Getting the Japanese fonts going only took a few minutes -- I'm not that Linux savvy. CTRL+Space to flip between Japanese and English.
I agree, the keyboard is small... they all basically are. The problem that I have, is a tendency to hit the trackpad with the side of my thumb while I'm typing, which sometimes changes the caret position.
I use it to study Japanese and type my Japanese journal entries for lang-8.