I'm not great at these time calculations, but I'll relay my experience and maybe it will help you figure it out.
I flew to Sydney last year from Washington, DC, and I think flight times would be about the same from NY to Auckland assuming you have a layover in LA or San Francisco. I had a 5-hour flight to LA, then a 15-hour flight from LA to Sydney. I left Washington at 5 p.m. on February 6 and arrived in Sydney at about 8 a.m. on February 8 (Aussie time). Basically, it's like you're traveling for 24 hours straight, but you lose a day by crossing the dateline.
So if you have a 2-night pre-cruise stay in Auckland that begins on February 1, you would have to leave NY on January 30 according to my calculations.
I'm not sure about VISA requirements for NZ, but for Australia all you need is something that I think was called an ETA (electronic travel authorizaton). You can get this through the airlines or a travel agent, as well as an Australian Embassy. The airline asked me for some personal info and a passport number, and my ETA was approved while I waited on the phone.
I visited Sydney, the Blue Mountains, Cairns and Port Douglas. If you need any specific info about these destinations, e-mail me at firstname.lastname@example.org
. You will love the land down under!