Free public schooling is only available to citizens and permanent residents.
Have a look at our Education and Schools in New Zealand to find out more.
I've made a few enquiries, and will let you know if I get any feedback on what you might expect to pay.
I got some updated information. Having spoken to an expat who stays there, it seems you should have access to free public schooling if you have a work visa.
I'd double check this with NZ Immigration to be sure, but there is hope.
Hope everything works out!