When you double-click a MID file and actually hear distinguishable music as a result, it’s because whatever your default MIDI file player is (QuickTime or Windows Media Player, usually) has access to a General MIDI sound set. See that link for more details, but basically, it’s a known set of instruments that are supposed to sound more or less the same across different synths or music programs.
The Halion Sonic plugin that comes with Cubase actually includes a full GM sound set. I haven’t done this in quite a while but I think I remember that if you import a MID file to Cubase, it puts each part on its own MIDI track and assigns each part with an appropriate Halion patch (assuming the file was saved with GM patch numbers).
I don’t know if that’s changed with the current Cubase, but if it still works, you’ll probably find that the sounds Cubase gives you are actually better (higher quality, more professional sounding) than the ones in your default MIDI player.
EDIT: Take a look at the “Importing MIDI Files” section of the Cubase operation manual.