There is a public shower landside, once you leave the secure area--it's to your immediate left, and you can hire a towel and toiletries from the convenience store near the McDonalds. They're not maintained to the same standard as an airline lounge, but they're more than adequate.

You can also prebook www.cheaptaxi.co.nz and pay less than half the metered charges from the airport to the city and vice versa. Use them all the time. No issues.

I always have deodorant in my carry-on.