Some games have an option on the main menu to view your product code. I'm not familiar with this particular game, so I can't comment on it. If not, I'm not sure there's a way to recover your product code. You could maybe try just copying over the game files from one computer to another, (or just burn the entire game folder onto one (or more) CD-Rs and copy the files to your new computer) and see if that works.
I remember Quake III Arena actually let you change your product code at any given time via the main menu. I always thought that was weird. I couldn't see the purpose of it, other than to encourage piracy.