I'm afraid the serial code pretty much equals the ownership of a game, nowadays. So basically, if you lose the code, you lose the game.
Since The First Decade is not a product on the Origin system, you can't have somehow registered it in your EA account, either. For a completely offline install, there's no way to ever have any kind of online recovery system, since there simply isn't any kind of online registration that stored it for you in the first place.
You could try contacting EA tech support through help.ea.com; perhaps they'll help you along if you can give them some kind of proof of purchase.