It sounds like you have (at least) two issues. 1) Your game should show up in iTunes after you bought it. 2) Your game is useless if you can't update your firmware.
I would focus on upgrading your firmware before you worry about your game. It won't play on the previous firmware from what I understand. If your library is backed up, maybe a full restore will do the trick, but I'm not sure. And of course you'll wipe your iPod clean and lose all of your settings.
Are you sure you downloaded the game when you bought it? Some items go into the "shopping cart" to be downloaded later, when you select them. Check your iTunes account for a shopping cart and also check your account info to see if your order is reflected there. If you're sure you've actually downloaded it, maybe try and locate the file in Spotlight just to make sure it ended up on your computer. If you paid for it and downloaded it, and it's still not on your computer, that's probably an issue for iTunes customer service.
