Well I am from NZ and spend plenty of time down there, I have a house in New Plymouth, Taranaki, North Island.
cost wise, pretty much from nothing on a do it yourself, free range to over $10000 for a high country station paid hunt.
it costs $25 at Auckland airport for a NZ police temp licence, Valid one year.
Likley $100 for shipping extra baggage fees, each way.
There are no tags, but a permit is required to carry a firearm on crown land, these are free, and are obtained form the department of conservation (DOC)
There are quite good heads 10-16 point 5x5-8x8 taken every year in back country NZ, they are normally hard won, threw lots hard work and sweet, Not all areas have quality gens in the heard, so if it a trophy your after research will be required to source likely areas,
I hunt the Ruahine, Kaweka ranges mostly, lots of deer but not over great red stag heads, ( mate calls them Ruahine rubbish)
lol, world class sika deer heads in Northern Ruahine and kawaka, these are a real fun deer to hunt, hard to put a hole in, but you get on to plenty in a day.
Cabins (huts) are common threw out the back country of NZ and normally cost $5 night good place to base oneself, fly camp away from hut for best results,
North island is mostly thick bush hunting and most shots will be inside 100m, take a compact rifle, South island is more open and long shots are likely 250m-plus,
If I am down there, when you are, I will be prepared to take you out, and hunt the kiwi way, I also have a mate who is a very keen hunter and does sum guiding, May be able to hook up something with him,
The roar is a busy time in the Scrub in NZ, esp the few weeks around Easter, Generally I don't hunt over Easter, and instead go fishing, Marlin, tuna, snapper etc, too many once a year bonehead hunters in the hills.
Rifle wise, the most common hunting caliber sold in NZ, is the 7-08, and esp in the north island its likely to be chopped to 16-18"and suppressed.
Take a fishing rod, pack light and make sure you fit, and you have a great experience.