Drinking water is going to be the least of your problems on the Aucklands. Here’s a paper on climate. There rainfall isn’t extreme (between about 60 and 80 inches depending on locality) but it rains between 311-331 days per year, mostly in the form of light rain and drizzle. As the photo I linked to shows, there are small streams everywhere.
However, if you do need to transport or store water, there are a few possibilities. There seem to be only about three or four species of tree found in the coastal forest in the Auckland. However, one of them is Pseudopanax simplex. The bark of a related species, P. arboreus, was used by the Maori to make small water vessels, and the gum was used to prevent leakage. Given time, you might make sturdier vessels by carving bowls out of rata.
Incidentally, the Southern rata has additional uses besides firewood. The bark was used as a medicinal to treat wounds, and the roots were made into fishhooks.