Try a mainstream Japanese vehicle with a proven service history. Avoid engines fitted with turbos, drivelines that incorporate all-wheel-drive systems and continuous variable transmissions (CVT).
Safety-wise a couple of airbags and anti-locking brakes (ABS) are going to be the best you can expect in this price range.
The budget:
$5000
The shortlist:
Toyota Corolla
This hatchback has the name for reliability in this price range, and we spotted a Corolla 1600 GL 1996 for under $5000. But the brand is very popular and New Zealand-new models are sought-after and quickly snapped up. And never assume the Corolla is bulletproof. It is important that you have it the car properly inspected before buying.
Nissan Pulsar
If you can't afford to pay for the badge like the Toyota you could probably pick up a later model Pulsar that has travelled less distance. We found a 2001 LX for sale for under your price. Reliability is the Pulsar's strong point The chain-driven engine has its benefits provided the vehicle has been regularly serviced.
Honda Civic
These cars are easy to maintain and and cheap to service with a late 90s model in your budget. But it is best to stick with a base model with a single-cam engine. This is a bread and butter vehicle for any local garage to look after.
Driven recommends
A pre-purchase inspection is highly recommended and make sure there are no problems with the engine, cooling system and transmission. If you can happily drive a manual then this also helps reduce future risk. Any of these vehicles will do the job.