Just got back from NZ two days ago. I flew Air New Zealand there and back.
You have to ring Air New Zealand before you leave and let them know you will be carrying a Firearm. They will ask:
1) How long is the gun case and is it locked ( as it is oversize freight).
2) Will you have more than 5kg of ammunition and is it seperate to the rifle
3) Will the bolt be out of the rifle (just put it next to the rifle in the case).
4) Your flight booking number.
5) The type of rifle it is and the action type eg Remington, bolt action.
6) Magazine capacity
Sporting goods on Air New Zealand attract half excess baggage fees so make sure you state they are sporting goods. NZ is very shooter friendly.
When I checked in, NZ check in staff asked if the ammo was in factory packaging ? (regulations state that the individual rounds must not contact each other) I use factory ammo boxes anyway so that there is no confusion. I also have a small lockable tin carry box in which I put the ammo, patches, scope mounting allen keys, collapsable cleaning rod etc.
You will need quite a bit of Australian Issued paperwork from customs to take your rifle and excess ammo home with you so make sure that is done. Australian Customs have always asked me to remove the bolt from the rifle while travelling.