In my 14 years of living in Dunedin on the S Island of NZ, I saw few people fly fishing for strictly marine species . Down there, substantial runs of sometimes very large sea run browns occur in estuaries on both the E and W coasts. Chinook are relatively restricted to a few river systems on the S Island. I've fished for kawahai, the so called sea trout which is not a salmonid from beaches and rocks - pound for pound, they are amazing - some spectacular sizzling runs and aerobatics. They are fairly common but travel in fast-moving schools, so better if you have a boat available. Offshore, barracouta (not my spelling mistake - a type of barracuda), tuna and mackerel and few other species are targeted with the fly. The continental shelf drops off fairly close to land, so tuna fishing, especially on the W coast is fairly inshore.The billfish sports fishery is a major one in the top half of the N Island. I'd guess that there are probably more saltwater fly fishing opportunities in the N Island, the north of which is sub-tropical. Certainly, saltwater fly fishing is much more developed in the US. That might have something to do with NZ's great wild trout fishery. Remember that it's only a 3+-hr flight to some great fishing opporunities in Vanuata, Tonga, Fiji etc.
Enjoy your time there
Graham