Unit K, 175 Rongotai Rd. Kilbirnie, Wellington, New Zealand
Foodville in Johnsonville
There is a little Korean shop on Cambridge Terrace in Wellington CBD which sometimes stocks some Japanese food, but mostly prepackaged stuff.
Opposite Pak N Save in Kilbirnie there is a Chinese supermarket which has basic Japanese cooking essentials.
Dixon Street in Wellington CBD has a Chinese supermarket as well which has frozen goods and a large range of Japanese snack foods as well as cooking essentials.
Yans Supermarket in Hopper Street (Wellington CBD fringe) has a small and expensive selection of things like mirin, sauces, sushi pockets, etc.
You can find a lot of the cooking basics in the "Asian" sections of Woolworths.
There is also a Korean shop in Wellington CBD on Dixon Street near the corner of Victoria Street which often has a few Japanese items as well - also they tend to have fresh mochi which is delicious... rather expensive there, probably due to location.