First of all, welcome to the forum and good luck with your move! Queenstown for the start of the ski-season - it should be glorious
We bought our first vehicle $1,500 from a friend over two installments, so I don't have first hand knowledge of this... However, I have done an extensive search, and I can't see anything that says that there is any limit on the cash you can use to buy a car (cash in hand may be a useful negotiating tool to bring down the cost of the car, too).
You do need to be aware that if you are bringing more than $10,000 in cash (i.e. not a bank to bank transfer) into the country, you will need to fill out a Border Cash Report:
I am not sure whether there are duties/taxes implications of this.
I've also pulled up a few sites which may be of help in respect to your rights when buying a new/used car:
Please do be aware of a few things with respect to cars in NZ:
You do not have to have insurance when driving. The implication of this is that there are lots of young kids (minimum driving age has only just been *raised* to 16...) who go out and buy their first car without putting insurance on it (and plenty of adults also who decide that they don't want the cost of insurance!) - if they prang you, you may not be able to claim back from their insurance company. So do make sure you are fully covered!
The WOF (Warrant of Fitness), the equivalent of the MOT, only lasts for 6 months and it is illegal to drive without a current one unless you are taking it to the garage to be certified.
The Reggo - car registration / car tax - can be bought for 3 months, 6 months or a year, (slightly more expensive to buy for shorter periods), however, unlike in the UK, if your car gets scrapped or certified unroadworthy before the end of the reggo period, you do not get any refund, so you need to consider if you are buying/running an old banger whether you think it will last the whole year!
The drink/drive limit is a lot higher than in the UK, and there is a much more relaxed drink/drive culture, which scares me a lot when I go out on a Friday or Saturday night - it is not uncommon to see cars weaving about. So, particularly on those narrow and windy South Island roads, be aware of other users and that they may be even more unpredictable than usual...
So - long answer to a simple question, but I hope I've helped out!