With that fare basis the booking class is K. Unfortunately K on Air NZ is non mileage earning, both with airpoints and with any other frequent flyer program.

IIRC K is the highest non earning class - it may be possible to pay a modest extra fare to get mileage (plus change fee). However you'd need availability in a cheapish booking class (eg Q, V or W or T) for this to be worthwhile. Unlikely this close to your trip.