There are no tags in NZ. You will need to get a temporary firearms licence from the police at the airport when you arrive cost is about NZ$25-50. Then you need to get a free hunting permit from the Department of Conservation and go hunting. A good area is Mt Cook village. I have seen and hunted tahr from the roads around there. You could camp in a few of the valleys or stay in Mt Cook village or there is a campground with cabins showers etc about 20k back down the road. Can't remember the name at the moment.
Also a few helicopters that could fly you to a few huts in the areas. The Murchison valley is popular for flying in to with helicopters.
Takes about 3 days to get a tahr. First day glass, glass and more glassing. They feed down in the late afternoon, stay low over nigh then climb back up in the morning. Once you've seen where they are coming down to climb up to there in the early afternoon. You usually won't get on to them in the morning as they will climb up faster than you can.