mad_atta and number_6 both said what I would have said. If you are coming in to AKL for only a few days Coromandel is the best bet to see nearby. If its not too cloudy the views from Sky Tower are great, and if you're that way inclined you have the option of climbing to the crow's nest and/or jumping off the tower!

OTOH, if you fly via CHC instead you would have enough time (just) to see some of the mountains and bush (rainforest) in the South Island with a short trip across Arthurs Pass to the West Coast. Weather could be iffy that time of year though. If driving there is a chance of ice and snow but probably okay as its not really cold by then. You can avoid this by taking the train to Greymouth (of course it limits the options on the west coast itself but as you dont have much time anyway it may not matter too much) - which is a very scenic ride across the plains, over mountains and through gorges.

Hope this helps!