Provided you abide by New Zealand's road rules, you should be able to travel safely, and without incident.
Occasionally, some mountain passes do close for short periods during winter, due to snow or ice. If that happens, you will simply have to park up until the road opens, or travel via an alternative route.
The key points to remember for driving safely in New Zealand are: We drive on the left (KEEP LEFT), and NEVER attempt to pass on Double Yellow Lines, or if the Yellow line is on your side of the road.