My (limited) experience is that you're gonna want to call on this one. If you have a trusted friend in the US, they can probably make the call -- as long as they have the PIN for your account, they should be able to make the change.

However, UA has toll-free numbers in a lot of countries. I'm in PEK right now, had to make a schedule change like you, and I just used the China toll-free number to do it. No drama at all. Have you checked to see if there's a local number for UA in NZ?

Skype is another fairly easy solution if you have decent Internet.