New Zealand is a small country. You will see a lot in that time. I agree with everyone so far. If you are arriving in Auckland, it would make sense to head north first. Northland has a lot to offer the traveller and photographer. Go up one side and down the other and make sure you get to see the very top, it is spectacular. Further south, the North Island West Coast is a bit of a hidden treasure. It has quite a coastline. There is also what is called "The Forgotten Highway", north of Stratford if you look at your map, which is a stunning drive.
Like everyone says, plan to spend a lot of time in the South Island, particularly the west and south. The hills are bigger, the vistas are grander and the roads are emptier! It is also more camper-van friendly.
I think your Sony will be just fine for the trip. I have found in other countries I need a 24mm or wider lens but, somehow, in NZ 28mm is just fine (in the old film SLR terms), so the R1 will work well. The wildlife here is quite elusive and what you can't catch with the 120mm end is still hard to catch with a 400mm!
Another tip, bring a tripod and umbrella. Yup, it rains a lot and light levels are often low. Be prepared for fairly drab colours. They work well with moody weather but can be disappointing in sunlight. The forest has very dark tones so underexposing 2/3 to one stop helps no end.
Hope you have a wonderful time.