From Zadar to Dubrovnik, there is a travel distance of about 350 km, which equals a bus travel time of 6-7 hours. Trough out the year there are direct buses, most in the summer season, where additional seasonal lines are operated. You can see timetable and buy ticket below. If you are doing a round trip you can save money if you buy a return ticket from same bus company. You can see timetable for buses from Dubrovnik to Zadar here.
It is not possible to travel from Zadar airport; from there you first have to travel to the main bus station in Zadar to catch the bus to Dubrovnik