Hi and welcome to AFF leeber
Airpoints works a bit differently to other frequent flyer programs. Instead of miles or points, the currency is airpoints dollars. Like the name suggests it is easiest to think of them as a $ substitute. Earning is based on routing, airline and class of travel (ie deep discount economy = no earning, discount economy = some, high economy = more, business = even more, first = more again). Spending differs between Air NZ flights and others. On Air NZ flights you use airpoints to pay fare as if they were aussie $ (this ignores special discounted awards if you have status which are fixed rate but only for business class). You can't pay part fare in cash (credit card) and part in airpoints dollars though, it is all one or the other. On partner (star alliance, plus Virgin Atlantic, Mexicana and a few others) flights, awards cost a fixed amount based on route, distance, and class of travel.
I presume you are already a member of airpoints? If not then you need to sign up to join, but this costs $50. Depending on your flying patterns (both this trip and future ones) it may be better to join and credit to a different program.