Jambo! New Zealand is a campervan and hiking-trail country, so offline is not an option. Install a JamboSim eSIM before you fly and land at Auckland already online.
Airport Wi-Fi works in the terminals, but data matters for campervan pickup and booking Great Walks.
The AirportLink bus plus train, or SkyDrive coaches, reach central Auckland in 45 to 60 minutes.
AKL has separate International and Domestic terminals with a free shuttle between them. Local time is NZST/NZDT.
SkyBus / trains
Yes, your phone connects to a New Zealand network as soon as you land.
Yes, coverage spans both islands wherever there is signal.
Install before departure and activate on arrival.