mostly for things like contact details, I suppose?
Yes, or a new passport number, or you get an extra qualification.
In this case, i'm guessing our only option is to continue with work visas until I have 3 years' experience in my field?
It sounds like it.
3 years working in NZ in an English-speaking environment would be more than enough for my partner to qualify in terms of English level (he has no English qualifications)?
Looking here
http://www.immigration.govt.nz/opsmanual/30541.htm at SM5.10 b ii, it seems that depends on the job he has being a skilled job.