You may have to change terminals, so use the info on the plane and ask the staff on the plan what gate you will be landing at. Make a little noise and they may help expedite you off the plane.
If they’ve sold you the tickets they need to get you there so they are motivated to put you in the seat booked and not put you up in a hotel until they put you on another seat on a different plane. Everyone will be working to get you there.
As as is good practice anyway, pack a change of clothes in your carry-on in case you are waiting for your bags to catch up with you at your destination.
Enjoy NZ.