Travelling by direct bus from Dubrovnik to Zadar is possible whole year around, during summer season, more departures is available. If none of departure times should be suitable for your itinerary, you could travel to Split and then change bus there. Below you can see the departure times from buses from Dubrovnik to Zadar. The travel time 6-8 hours depending on travel route of the bus.
If you plan also to travel back to Dubrovnik, you can save money by buying a return ticket with the SAME bus company, you can see timetable for buses from Zadar to Dubrovnik here: