I think you first need to decide if you are going to hunt with a guide or not. If you are doing it all yourself then you need to sort out these logistics:
Check with your airline that they permit firearms.
Get a visitors firearm permit.
Search DOC (who administer public land) website and find a few hunting spots (suggest middle to lower South Island).
Get a DOC hunting permit.
Also while I haven't been to Germany, I suspect that hunting is very different here. You'll need to find a hunting area that matches your fitness and confident in the mountains. You should consider contacting a helicopter company and asking them about hunting spots they recommend. Helicopters are a fairly common why to access the backcountry here.
If you want to call me and discuss your trip send me a PM and I'll share my contact details. I have hunted overseas a few times.
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