Here you go!
I used Shazam app and detect the song name, it's quite popular so Shazam app can easily detect it.
It's called "Merry Christmas Mr. Lawrence" by Ryuuichi Sakamoto, the title track to the movie of the same name. The movie was originally called 戦場のメリークリスマス (Senjō no Merī Kurisumasu -- LIT: Merry Christmas on the Battlefield) in Japanese.
This one happens to be the Piano version, and the clip from the cat video is the intro to the song.
Here's the original from the movie:
Kinda gives off a CLANNAD feel to it, doesn't it? ^_~
Also, in case you're interested... There was a song that sampled the original and it was used in a Dance/Electronic song called "Heart of Asia" by Watergate. (The chorus)