The Verdict: Need More Info
Your partner can sponsor you only if they are an eligible New Zealand citizen, which is a defined status. Check their status through the Home Affairs visa entitlement verification service (VEVO).
A New Zealand citizen on a Special Category visa generally counts as eligible if they were in Australia on 26 February 2001, or for at least twelve months in the two years before that date. Many who arrived later do not.
There is a second way. Since 1 July 2023, New Zealand citizens on a Special Category visa can apply for Australian citizenship directly. A partner who has lived in Australia for four years may be able to become a citizen first and then sponsor you.
Once your partner is an eligible New Zealand citizen, a permanent resident or a citizen, the partner visa runs on ordinary terms.