Once you are in NZ with your rifle you can hunt in any of the Department of Conservation (DoC) land by getting a permit. For Arthurs Pass any DoC office in Canterbury shouldbe able to issue the permit. There is no charge, but they do want to sight a firearms licence, not sure what you do there, show your aussie? or the import permit for your rifle.
Only problem with hunting on DoC land is animal numbers are variable and a bit of local knowledge goes a long way. Using a guide would be better, if you can afford it! Most guides in NZ hunt on private land although some will take you on to DoC land.
The DoC website has some good info for hunting around Canterbury
Hunting in the Canterbury region