Welcome to the forums.
Truck shipping is usually around $2/loaded mile, plus extras.
An older Airstream (pre-1969) will fit in a container, you may have to remove the air conditioner from the roof, and flatten the tires.
1969 and newer, you will have to get measuerments of the trailer, and check inside measurements of the container to be sure. Do you know if there are any RoRo facilities in New Zealand that can accomodate a trailer?
There are several forums members that have shipped Airstreams, either in conatiners, or RoRo, if you type in "overseas shipping" in the search area in the upper right hand corner of this page, you will be able to get more information.