Check your console's storage device:
XBOX 360 Dashboard > Settings > System > Storage > Hard Drive > Games and Apps > Battlefield 3
You should see Battlefield 3 HD Content (if you chose to install that), Battlefield 3 Title Update (183 MB), Battlefield 3 Update (2.1 GB), Battlefield 3 Premium (56 KB), Multiplayer Update 3 (2.4 GB), Multiplayer Update 4 (1.5 GB), and Online Pass (56 KB), Armored Kill (1.6 GB).
Like I said, you don't get the option to download Close Quarters as it does cost 1,200 Microsoft Points alone, you need to have Multiplayer Update 3 downloaded correctly in order to access close quarter servers. Battlelog will of course state you have access to BK, CQ and AK as soon as you purchased Premium but you still need to download the multiplayer updates.