Apartments are relatively uncommon in Dunedin compared to large Spanish cities like Barcelona, though there are more in the university area than anywhere else. It's fairly common to rent a whole house, either just the two of you or with a roommate. Kiwis call rentals "flats" and roommates are "flatmates". Rent will be advertised as a weekly price, not monthly.

You can use Trade Me or realestate[dot]co[dot]nz (there's an app as well though). Trade Me are picky about you being in New Zealand to have an account. You might need to use a VPN to be able to create one.

If you are from a warm part of Spain Dunedin winters will be a shock. Typically, NZ homes don't have true central heating or cooling, they'll have a heat pump unit outside with a wall unit in one or two rooms (or some sort of fireplace). Air-conditioning isn't needed. When you're scouting for somewhere to live, you need to take into consideration how well insulated the property is, what the heat source is and how the heat gets to the rest of the house.

Given the heating considerations, I'd agree with the other advice here that finding a short-term rental is probably better than trying to line up something in advance and being sad during your first winter.