by january do you mean july?
Shouldn't be too hard to find somewhere to rent at that time of year, probably your best option would be to budget staying at a backpackers / hostel for 2 or 3 weeks when you arrive. That way you can get a feel for the place and find somewhere to rent.
Usually someone will take a lease out on a house and then advertise for flatmates for the left over rooms. You could probably expect to pay something between 100-200 depending on number of people and quality of house etc. Obviously you could spend more if you wanted or possibly less if you're sharing rooms with other people.
www.trademe.co.nz is a good place to get a feel for what the rental market is like -
http://tinyurl.com/wanakarentals