Its possible, probably. Look at flights and such, and pack a tent. Finding lodging at a ski area in NZ might be hard to do cheaply. Hostels wont be close to mountians either, so... its looking iffy for under 2000. Good luck however.
Alternatively, you can just fly to a volcanic peak, like Mt Hood, Mt Baker or Mt Rainier (all local ones for me), get lodging in a nearby town or city and ride for a week using a rental car. Hood will be the only one with a lift and a park, but making features can be fun, if you take the time and put some real effort into it. You could also just road trip it, if you are close enough.
I'm here now and my plane ticket cost $1,200 from Reno round trip, you can stay at 'Backpackers' which are like hostles close to the mountains for 20nz which is $15US. The buses to the mountains cost 20nz but the people here will pick you up if you hitch a ride. It has been snowing for the last 3-4 days and all the mountains in NZ are closed b/c of the weather, so their winter is going really well! I've been in Christchurch for the last 3 weeks but i'm going to Wanaka this weekend if you have any questions.
CU-Boulder Class of 2010
"Over time, most people experience life involving love, suffering, compassion and an unspeakable drive for something new...For me there's skiing, nothing more nothing less and it encompasses everything, everyday I'm out there."
- Pep Fujas
look for flights to nz from different cities closer to nz and then ride the train to that city. last year i was trying to get out to tahoe from utah and a train ticket was like 65 bucks. that's my suggestion. then camp