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.