I’ve tried to reply to some DMs, but I can’t, so I’ll post this here as a general guide for what to expect in NZ.
First of all, I think there’s a common misconception about how small/large NZ is.
If you are coming for one month, I would recommend that you only visit the South Island, and you be prepared to maybe hunt two species (Red deer and Chamois for example).
Unless you are hunting high fence game parks and paying thousands of dollars, hunting even one species takes a lot of time, here’s why;
During the April/May season, the weather is extremely changeable, you will only get small windows in which the weather is actually suitable and safe for being in the mountains. For example, in the month just been, I had a party over from Australia. Out of the 12 days they were here, we only spent 7 in the back country due to weather.
The species generally do not cohabitate, with the exception of Red deer and chamois. This means that in order to get an alpine triple (red deer, chamois and tahr) you need to hunt a minimum of two spots.
There is an abundance of game, but, representative trophies are hard to come by and generally located in areas of extreme remoteness. I would advise against helicopter trips into the back country. Most of the areas accessible by helicopter generally get very high hunter traffic in Autumn due to the red deer roar (or rut). Most, if not all, helicopter pilots in NZ are engaged in wild animal recovery and are shooting and recovering red deer for sale on the commercial market. This means that most areas which are near to helicopter hangers (and some which aren’t) generally hold much lower animal numbers.
I would recommend that you be prepared to come here and hunt with a higher than average level of physical fitness. The mountains are deceptively difficult, as most of the best hunting country is glaciated valleys near to the main divide of the South Island.
Can you bring a bow? Yes, you can import a bow, arrows and broad heads with no permit or paperwork required. Would I recommend bow hunting on your first trip here? No. I would recommend that unless you are hunting high fence/game parks, you bring a rifle. I would also recommend something on the lighter end of the spectrum. For example, I hunt with a Forbes 24b in 6.5-284n and have shot all game species in New Zealand with this. You need a light rifle which you can comfortably shoot from less than conventional positions, and out to about 500 yards. Think about shooting off a ridge which is at about a 30 degree angle.
Key pieces of equipment for representative trophy hunting would be;
- 3 season tent (something light, most serious hunters here use the MSR access, Mont Moondance or Mont/Hilliberg tunnel tents.
- spotting scope with phone scope adapter, you want to be able to get good imagery of an animal and make a good assessment, as some places you cannot walk out the way that you walked in.
- adequate time, be prepared to be grounded due to weather, and make shorter trips to more areas during your time here.
I would recommend flying to and basing yourself out of Christchurch. Despite the fact that many foreigners base themselves out of Queenstown due to the fact that it “looks nice” Queenstown has a lack of accessible public land for hunting.
is yours today for your excellent pix and joining from NZ. 
I can give you some upfront advice before you book flights etc.
