Gday Joseph,
I was in Queenstown last winter, and after fracturing my wrist on the first day of snowboarding

I suddenly had some free time on my hands (no pun intended...)
The airport itself is relatively easy to photograph at, but there are few movements, and the majority of those are sightseeing cessnas and helicopters. Kawarau Rd to the south of the airport is quite good, and from there you can see aircraft lining up (if taking off to the north) and climbing out over you to the south (it's kind of like the Qantas Drive of Queenstown). I was staying on Frankton Rd not far from the airport, and right on the water - if you walk back towards Queenstown along the water's edge, there are nice shots to be had of aircraft climbing out with (I think) the Remarkables in the background. Also, the walk provides amazing scenery photos too!
Also, the roads around the outside of the airport are close to the fenceline, and to the east of the airport there is a shopping centre where you can park and walk back to have a look around. I was up at the fenceline for some time with my camera and never had any problems with security.
Hope you enjoy the trip, and if you go snowboarding, invest in some wrist guards