There will be ticket-vending machines at the airport, probably right near where you board the bus. The machines take credit cards (not Amex). There should be an employee selling tickets (for cash) near the bus boarding point, and the website says you can also pay the driver in cash. The policy is not to give more than 20 euros in change, so you don't want to be stuck paying cash with only a 50-euro bill in your hand.
I don't remember whether I took my suitcase (somewhat larger than carry-on size) onto the bus and stored it on a rack or whether I was directed to put in the hold under the bus. Barcelona has a lot of highly-skilled pickpockets. They are not usually interested in your luggage. The main thing is to guard purses, wallets, and electronics. Now, if you are traveling with a $1000 suitcase (believe me, they would know) and/or they see you place $$$$ electronic gear inside the bag before you stow it, that might be quite a temptation.
The Aerobus doesn't make many stops. I believe it is direct to Pl. de Catalunya without any stops along the way. That's usually not the sort of environment a pickpocket prefers. They like the hustle and bustle of people getting off and on the bus so they can pick their opportunity and disappear quickly with the loot; they are usually long gone (off the bus) by the time the victim realizes something is missing.