Yes, you can apply from overseas. Use this INZ page
http://onlineservices.immigration.go...or.htm?level=1 to find out which office you should apply to. You should look at the INZ fact sheet about this visa
https://www.immigration.govt.nz/new-...-resident-visa to make sure you meet all the requirements. We know from experience that INZ take seriously applications when people can show around 3 months' evidence of living together. See old threads
https://www.google.co.uk/#q=site:enz...tnership+proof for discussion and examples of what evidence can be provided.
The application would be processed in one of the INZ offices. It will not necessarily be the nearest one - INZ move work around depending on what offices are busiest at the time. If and when the CO wants to interview one or other partner, they will arrange it individually, sometimes as a phone call and sometimes by meeting in person.
You would not be able to travel until and unless you get issued with a visa. (You would not even be allowed onto the plane at the foreign airport without a valid visa.) So no, Immigration at the airport on arrival in NZ are NOT the people who decide such cases. They have the power to turn someone away if there is something found to be illegal about their situation, but that should not be the case for you once you have gone through the full processing of a visa application.