Auckland to Hamilton (State Highway 1) is relatively simple - flat, 2 lanes each way for the most part. Everywhere else will be undulating, bendy, 1 lane each way, with the occasional passing lane. The big mountains are south of Taupo. Road condition is generally OK to good.

There shouldn't be a problem with weather at that time of year. Fog may be an issue, but it's not a big deal. It doesn't snow there, but it may rain.

It's a fair bit of driving for such a short period of time though...? Is Tauranga essential? I don't think there's much to do there during winter...