You can get a NZ Hunt license, and a Rifle License to bring your rifle on the internet. That's the easy bit. Here is some info from my experience...
1/ DYI Wilderness: A lot of flat footed Aussies fly over for DIY hunts, and unless we train very very hard its a disaster. Most need 2 or 3 trips to get a nice one. DIY involves booking a hire car, leave it at a Heli pad, fly up with your gear and supplies, start glassing, and get lucky... costs might be $2800 to $3500 for 2x depending on the Heli distance/time. You will need the right gear should you be snowed it, your buddy must be bush and mountain smart, survival knowledge, super fit, and it will be a big learning curve. Plus you both might not get anything... yes you will see Tahr, but they are damn hard to get close to, and that can lead to taking risks and safety concerns. Plus if one person looses his will to be in such a harsh difficult environment and physical stress, then its a DISASTER as you should always hunt together. If the other guy says stuff this, has sore legs, or blisters, or has a fall and damaged knee or pulled muscles, and decides to sit in camp, then it is unwise to then go out by yourself. All the young local Kiwi hunters are super fit, and go with a well matched mate, with equal stamina. Half a day of untrained unfit legs can cause them to feel like jelly and painful, and not work on day 2... then the brain can crack, and its a mental problem then. We also had a big high wind storm and luckily our 1x cheap, and 1x 3rating tents held together. I know this can be a real problem, as it happened to me, we heli'd out early and thus my/our hunt finished before success. However DIY has its own special memories and achievements.
2/ Many of the main hunt operators do 1 day Tahr heli hunts, which are nice if your over 50 and unfit for severe mountain hunting. Not for me... yet..!!
3/ Guided "True Wilderness" hunt costs much more, and because you both are paying, you train bloody hard. If you can save up for a wilderness hunt, then thats a great experience, as the guide will know the area, he will supply all the right camp gear 4 rated, radio and satelite phone, etc etc, and probably a 95% success rate. Heli up, then 5 or 6 real hunting days, then heli out.. This will be one of your most demanding and rewarding hunts ever. If you can save $5000 or $6000 for 1x1, or maybe $4000each for 2x1 with a mate of same abilities, I'm sure you will get a good deal from a strong wilderness guide, who provides all references. There are Ranch, Private wild, Estate, Wilderness... all sorts of ways it is described and percieved. Make sure you get the hunt you want.
Season - Early winter April, May, June, July is the preferred time for nice coats, but still has a risk of being snowed in, and time lost, or late out. I'm a 54y/o warm blooded Aussie so Summer Nov, Dec, Jan, Feb is much nicer for 'soft' me, with minor chance of snow or storm, however I'm only after a skull mount, or summer coat.
CANT WAIT TO GET BACK...!!!
Pic are near MtCook... Weatherby Ultralight was long, 1.2" x3 grouping 100yds and not light, later S old it for a Tikka T3 superlight 300WSM, same hole at 100, apple" at 500yds..!!. Other pic a young Tahr.