If it is true that there is an agreement and law between New Zealand and Saudi Arabia of free travel then in this case you may travel under this law by your passport to Saudi Arabia without a visa. And then you can go the Embassy and get visa for umrah and thereafter you may perform it. It is not necessary to make the intention from your own country, but it is not lawful to cross the miqat without ihram.
Allah knows Best!
Darul Uloom Deoband