I made a page that does exactly that. It can also take costs for accommodation, car rental and daily food and entertainment budget into account.
For the trip you mentioned it might cost you a bit more than NZD 300.
Keep in mind that it will cost you an additional ~NZD 200 your you want to take your own car with you from north to south island on the interislander ferry. If you rent a car, it would be better to drop it of on one side and hire another car on the other island, pretty sure the rental companies have some sort of arrangement for that.
Fuel prices right now are $1.80 (Auckland CBD) or $1.70 (Auckland south) for 91 unleaded, but the pricewatch site gives you a better indication of fuel prices across all of New Zealand.
Your fuel consumption depends on a number of factors such as type of car, extra load, speed, terrain (flat vs. hills and mountains) your personal driving style, city vs. highway so it's really hard to come up with an exact number.
If you have 2 people in a mid-sized car with lots of luggage I would estimate 9 l/100km, and everything below that is a bonus. The fuel consumption car makers advertise are usually done under ideal conditions, and the real consumption might be up to 20% higher during everyday trips.