The town of Zadar has an airport (IATA: ZAD, ICAO: LDZD) located 8km from the city. It is a very small airport, and you generally won't spend much time in customs or passport control. The destinations change every year and vary depending on the season (See here for up to date details). In 2014 nearly half a million passengers passed through the airport (up from 272,675 in 2010).
The official website for the Zadar airport can be found at www.zadar-airport.hr/en.
There are busses that run from outside the airport and go straight into town .
The busses cost 25kuna. They can get quite packed after a couple of large planes arrive.
There are three stops - The airport, The bus station, and a bus stop in old town. It takes about 20-25 min to get from Old town/bus station to the airport.
Taxis should cost around 150kuna.