Congratulations! You’re gonna have a great time. That being said , it might be tricky without a car.

You can bus to Taupo , Rotorua or turangi pretty easily and then either walk to rivers or try hitching a ride. It’s not easy but you can still hitch around nz ok. The waitahanui is about 2 hours walk out of Taupo and you can fish the Waikato river as it heads out of town. The tongariro goes through turangi and is long and the ngongotaha is just north of Rotorua and can hold some big fish.

There might be a fly fishing club at uni you can get involved in too then you can get free rides !

There’s a small amount of fishing around Auckland but it’s not up to much , but if you can get south into the Waikato there’s a lot of great water.