Go into your ps3 Internet browser and enable cookies by pressing triangle, selecting 'tools' from the side menu, and enabling cookies.

You got a physical ps3 copy of the game. Inside should have been a 12 digit code xxxx-xxxx-xxxx to authorize your psn to play. Did you purchase the game new? If so the code will be included in the packaging.

I'm confused because in your original post you mention redeeming this code through the psn store but in a follow up comment say the code is not included.

You can check if your Psn is authorized by going to 'playstation network' >account management > transaction history > service accounts. Select service accounts and you should see ffxiv listed there. Select it and it should have the various game authorizations and when they expire. If it is not listed there then your code was not redeemed properly.

You do not need to re-download the game.