You'll be a lot better off with your stuff in panniers.
Assuming 350-400 miles in 5-7 days, you'll be doing an average of between 50 and 80 miles a day. You should try to ride that distance in a day sometime before your trip to see what it's like.
Unless you intend to actually tour off-road, get tyres that are suited for touring on tarmac.
Practical repair experience is kind of important. Try doing different repairs on your bike as it is now, even if it isn't broken. Then you'll know what to do if something breaks.
Let's see. You're out of shape; you've never ridden your bike all day; and you want to ride 50 to 80 miles a day on knobbies with a pack on your back - through the night, in the winter, on narrow roads.
If your bike can take a rack, fit one.
You can strap a light backpack or duffel to the top.
Low end MTB from respected brands are OK but discount store models can be dubious.
I did meet a very big, tall NZ traveller on a discount store bike with an army kit bag slung across his cheap rack. He rode across the UK a few hundred miles. It can be done but it helps if you meet up with a decent mechanic to straighten your bike out halfway through.
My NZ friends tell me that NZ is cooler and wetter than the part of Australia where I live, and I live in a cool, wet part of Australia (during the autumn, winter, and spring). Don't be surprised if you encounter temps near freezing during the nights in some places ... and of course, lots of rain.
Have you ever done a 50 mile day ride with a full backpack? Go do it ... and then decide if you want to go that route or if you would rather go with panniers.
EDIT: There's the darkness factor too ... because it is winter down in this part of the world, the days are very short.
Sounds pretty miserable. Is there a particular reason you have to go in June? Why not wait until spring (NZ), like October or November, and go then? That will give you time to get better conditioned, to do a couple of practice weekends to sort out the equipment, packing tools, etc. and you'd have the whole summer to do it.
I would not feel safe riding through the night without a good set of lights and lots of reflectors on my bike.
Definitely get panniers or a big seat bag (like a Carradice Camper), or at least strap the backpack to the luggage carrier of your bike. Cycling any kind of distance with a pack on your back is not fun.
Switch to slick tires.
At a bare minimum, practice fixing a flat tire.
The daily mileage is certainly ambitious for an out of shape, first time tourist on marginal equipment. But what the heck, have fun.
Yeah, bad idea. First tour should be low miles, safe roads, nice weather. And forget the backpack - miserable!!!
RE: the road - From wikipedia: "The road between Rangipo (10 km south of Turangi) and Waiouru is commonly known as the Desert Road, and is frequently closed by snow during winter."
Are you in NZ already? It sounds like maybe not. If you are in the northern hemisphere, I would respectfully and very strongly suggest you try something here, in summer, with nice weather and long days. If you are in NZ, wait for spring/summer.
If you're just going to be in NZ anyway and looking to tag on a bike tour - I think you should reconsider, respect the conditions that exist rather than trying to mentally force them to work out. Just go tour by car, and go back another time.
There's a great set of books - Pedaller's Paradise - one for each island, if you want to do more research. These have some info about road conditions, grades & services. I found them very useful.