Assuming you want to cancel your Air New Zealand flight. Then, at that point, there are a couple of ways of making it happen. You can either call the airline’s customer service line or you can go on the web and cancel your booking. Assuming that you cancel your flight on the online. Then you'll have to have your booking reservation number convenient. How to cancel Air New Zealand flight online? Read or Dial-+0064-9-357-3000 or OTA: +1-888-894-2072.