The simplest connection is to fly to Zadar and connect by ferry from Zadar to Preko on Ugljan island. The car ferry (line 409) from the Gazenica ferry port in Zadar runs almost hourly during peak season and the crossing is 25 minutes. The cost for a car and four adults is approximately €24. Timetable on the Jadrolinija website. There is a passenger only service, also from Jadrolinija, running from the harbour in Zadar old town city centre to Preko on Ugljan. The cost is around €2.5 per person one way. The Gazenica port is about 4 kilometres south of Zadar city centre.
The airport bus departs from outside the terminal building at Zadar airport and stops at Gazenica ferry fort, the main bus station in Zadar and in the old town city centre. Buses depart after every flight arrival. The bus is 25 kuna (€3.35) per person. For a taxi to the city centre expect to pay around €20
Another option is to fly to Split. The road journey from Split Airport to Zadar ferry port is 2 hours with a scenic motorway much of the way.