Can I take a bus from the airport to Zaton?
I don't believe there is a direct bus. You'd need to take a bus into town and then catch another bus on to Zaton. Some basic searches on buscroatia.com show only one bus per day leaving Dubrovnik and stopping at Zaton, so it looks like you will need to find an alternative way to get there.
Take the airport bus from the airport to Dubrovnik main bus station in Gruz. Bus is operated by Atlas, and you buy tickets in the arrivals hall. You can catch one of the Dubrovnik suburban buses - the operator is Libertas. You could catch bus 12, 26, 36 or 15, but ask at the bus station and there may be other options, such as a long distance bus dropping you in Zaton.
Suburban timetable is at http://libertasdubrovnik.com/wp-content/uploads/prigradski-zapad.pdf