Welcome and good to see you are planning well in advance. 4-5 weeks should give you plenty of time to explore and get in a number of hunts.
At this stage there are no resident vs. non-resident hunting regulations on Public Land. You are probably aware of the current Government's mantra to eradicate all introduced "pests" and that now includes some majestic animals which were bought to NZ under specific programs over 100 years ago to provide recreational hunting opportunities.
Yes, we are nation like many others, hell bent on trying to rewrite history rather than making the effort to learn from past experiences.
Unfortunately, that list of "pests" includes Tahr so you should find plenty of information on the Department of Conservation (DoC) web site.
Reach out if you are looking for any more information. I have a US VoIP phone at home - equivalent to a US landline if it's easier to have a chat.
I normally spend about 3 months traveling in the Western US during the NZ winter but those plans are on hold for now.
Stay safe and healthy.
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