Interesting idea! I have a few thoughts, in no particular order:
Getting There
This is, of course, the biggest challenge. And to be honest, it's a massive difficulty. A successful colonisation across 2000 kilometres of open ocean is a major undertaking. It requires a very good technological package.
While we don't know exactly what level of seafaring technology the proto-Aboriginal peoples of Australia had, it certainly seems to have been worse than that developed by, say, the Polynesians. I can't think of any equivalent island which was colonised until at least a neolithic level of technology. Even Madagascar - which for comparison, is only about 500km from Africa - went uncolonised until Austronesian and Arabic technology.
Still, it's not entirely impossible for some ship - or better yet, a whole fishing fleet - to be swept out to sea by winds/squalls/sudden storms/whatever and wrecked someone on Aotearoa. And I usually figure that any WI gets at least one gimme.
But remember that whatever happens will be a one-way trip, and it will be a very small starting population base. You'll be looking at some serious founder effects and inbreeding in whatever population emerges. And they probably won't be bringing a great deal of technology with them.
Also, while this is largely tangential, I doubt that the colonisers would come from Tasmania. We don't know exactly how much ocean currents changed during the ice ages, but I think that an equivalent to the branch of the East Australian Current that moves east from the New South Wales coast is more likely.
Effects on Native Flora and Fauna
These can be summed up as "moas are gone". In a very short timeframe, probably. When landing on an alien land with no knowledge of the flora or fauna, the obvious target for food is big game. Their ancestors in Australia have just finished wiping out the megafauna in a very short period; the megafauna will follow.
Much of the smaller fauna will survive; I doubt that these colonists have brought rats with them, and they certainly don't have any other domestic animals. Nor are they likely to find any.
Eruptions of Lake Taupo
The effects of this will be very very bad, but a civilization-killer it is not. Anyone who's within range of the (rather large) pyroclastic flows is dead. Anyone who's living under the ashfall will also be in a lot of trouble, but the ashfalls will (mostly) spread with the wind, and the people who live upwind will mostly do okay. It's significant that the Taupo eruptions were not (as far as I know) associated with any extinctions. In the most recent major eruption, even the North Island subspecies of various flora survived. If they can do it, so can humans.
In the long run, the ash will actually be a major boost to soil fertility.
Agriculture: Yea or Nay?
This is the $64 million dollar question.
Unfortunately, the short version is "probably not".
The reason for this is that the Maori, who had agriculture, did not find much if anything in the way of domesticable crops there. There were a few plants which they cultivated or just harvested for food, perhaps most notably bracken fern (Pteridium esculentum) and the
cabbage tree (Cordyline australis). But they didn't turn up much.
Of course, this is not a complete answer. The Maori were only there for a few centuries, and it does take time to get to know all of the flora, and even longer to start some plants down the road to domestication. But then the Maori had the advantage that they were already bringing agriculture in, so that gave them a rather large head start in terms of domesticating crops. Without that, then agriculture needs to arise
de novo - and that is a much harder proposition.
I'm not familiar with any New Zealand flora which could serve as a full set of founder crops. Bracken fern is not ideal, nor is the cabbage tree.
Offhand, I know of only two plants in New Zealand which can reasonably be considered for part of a founding agricultural package - and they're not enough. It has a relative of murnong (called
Microseris scapigera) which could probably be turned into a tuber crop - but it won't be enough in itself. The other is Warrigal greens/New Zealand spinach (
Tetragonia tetragonioides) which makes a decent leaf vegetable, but not much more.
What is needed is a few plants which are rich sources of carbohydrates and (ideally) protein as well. That basically means cereals and legumes, or a decent root vegetable if one's available (murnong is the best I know of, though).
New Zealand's legumes are mostly trees or shrubs, and I don't know of any which are particularly suitable. There's the scree pea (
Montigena novae-zealandiae, formerly known as
Swaisona), but that has the problems of being (i) a perennial shrub and (ii) growing native only in restricted areas on mountain slopes. There's a whole bunch of New Zealand brooms (
Carmichaelia), but again they're mostly trees and don't seem to have particularly accessible seeds.
I know next to nothing of New Zealand's native grass species, and those few I do know of are not suitable. It may be worth having a rummage around on google if any of them are useful (the Genocide does at least have a list of them
here).
So, all in all, while I wouldn't completely rule out indigenous New Zealand agriculture, it's not an easy proposition.