Q:

iTunes can't download song purchased from old computer, on to new computer

I bought and downloaded a song in iTunes on a PC of mine.  I never synced it to my iPhone.  Well, I got a new computer, and no long have the old one.  I think there might be a problem now. 
Here is what happens when I click Check For Available Downloads in iTunes:

And here is what happens when I click on buy, for the song in the iTunes Store:

Is this a case of a badly worded dialog?  I've seen something like this before in the App Store when I needed to reinstall a paid app.  In that case I "bought" the app again, the second time without being charged. 

A:

Unfortunately this is the expected behaviour.  Re-download of music is not supported. 
You have a only a few options:

Wait for iCloud to become available for Japan.  Currently Automatic Downloads for Music is only available in the US.  Once it is eneblaed internationally you can enable it as Hand-E-Food described in his answer. 
E-Mail Apple support and tell them you lost the music due to an hardware-crash or something similar.  There are reports that Apple support allows re-downloads of your complete purchase history.  But this is a one time offer and a courtesy of Apple.