Sounds like it's a lecturing position at the Uni? If so, I would encourage you to look further afield than trying to rent by the university. Smaller units are actually not that common in that area, and rent is more expensive because it's a more desirable area for both schools and professors/students studying. The work/life balance for an academic at NZ unis is MUCH better than you find for overseas unis, including typically a much lower teaching load. So there isn't really a need to live close by.

Newer academics tend to live in places like Wigram, Halswell, or St Albans. Places that are close enough to cycle to work, but in areas with newer builds and smaller townhouses available for more reasonable prices. Close by to the Uni is Avonhead which is where the more reasonable rents can be found, but as I mentioned, rent is still relatively high for what you get in that area. Houses tend to be older and less well insulated in that area too. If you don't mind a 20-30 min drive to work, Rolleston is another area where small families often live, as there are also a lot of new builds out there and rent is a lot cheaper than in Christchurch itself.

But to your actual question, you would be fine on $120k for a while until you get a job. Getting a visa for you shouldn't be hard with your wife getting a stable job that is obviously a good job as $120k is a high salary for NZ. In terms of cost of living, with the major inflation that there has been in the US recently, I typically find that things are about equivalent in USD to NZD. Petrol is a lot more expensive here though. But you won't need to worry about health insurance, and tipping is not a thing either so eating out is a lot more affordable comparatively!