Taxi to Airport in Dubrovnik

We had the hotel get us a taxi. We arranged this the night before our flight.

The trip is about 20 minutes by taxi. Just talk to the people at the front desk of the hotel and they should be able to arrange something for you and also suggest when you should leave to make your 7:30 flight. In Zagreb, our hotel was associated with specific taxi services and so could quote a fixed rate on the airport transfer. I assume that it is similar elsewhere in Croatia.

We made an arrangement with a taxi person at the stop outside the old city walls the day prior to our departure. Were picked up at 4am the next morning for an early flight, no prob.

My recommendation is definitely to hire taxi service that will give you a transfer to the airport. It is better to reserve them in advance, as there is a huge demand for them during the night and morning hours, given that most young people in Croatia goes out then. Taxi will pick you up and bring you to the correct destination, without having to manage on your own. You should search for online taxi agencies that exist in the Dubrovnik area and choose the one that suits you best.

Thank you for your advice about hiring a taxi the day before.

The shuttle bus for the airport leaves right in front of the Cable car station every 30 minutes and takes about 20-25 minutes. You can probably catch the 0533 bus and have an hour and a half for check-in. I'm pretty sure it cost us 100Kuna each.
There's a store right across the street next to a bakery that has the daily schedule and sells the tickets.