You don't actually need to play through the levels in order to get this achievement. Simply pick;
Local game
The Arena
Vs
Community Theater
Get New User-Created Playlists
Then choose the top game and press A, ready up. As soon as the game loads and you can see the doors to the lvls you have dl'd you can press start, then pick change arena mode. This will take you to the voting screen, vote, then repeat the process with 9 different lvls