port of stockholm-freeport cruise terminal

I'm assuming that "freeport" is the English translation of the port called Frihamnen. You mentioned that you did not want to take public transportation and wanted a taxi. There are three reliable taxi companies. They are Taxi Stockholm, Taxi Kurir, and Taxi 020. I usually use Taxi Stockholm. www.taxistockholm.se/english/. It will probably cost about 600 SEK to take a taxi to the airport from there.

Have you checked with Azamara? I would expect them to offer coach transport to the airport at a reasonable price. Or you might be able to share a ride with fellow passengers. PS - I was on the Quest last year and loved it.