Hi Jan,
Being an international traveller myself, I completely appreciate how much easier it is to do things on-line as opposed to having to make international telephone calls.
Here in the UK it is possible to enter and/or update your
Disney's Magical Express information on-line by visiting the following web page:- https://disneyworld.disney.go.com/wdw/reservations/dme/index?bhcp=1
I cannot see any reason why Guests from New Zealand would not be able to use this same form.
If you do encounter any problems, I would recommend firstly contacting the Company with whom you booked your holiday package with and asking them to update your flight details.
As a last resort, you could ring Disney's Magical Express direct by calling 001-866-599-0951. If you do end up going down this route, in order to make the call as economical as possible, I would highly recommend using Skype.
I hope that this information has been of help and wish you a magical trip!