As per the above comments, no way you’ll see all that in 9 days and you sure can’t do it on public transport.
My stong advice is hire a car, it’s very easy. My first time on the opposite side of the road was from LAX in full on rush hour. NZ is nothing like this and the Sth Island is empty. Just take it slow and in a few hours it’ll be fine.
In 9 days do Christchurch, Queenstown/Wanaka/Arrowtown, Te Anau, Milford Sound, Dunedin, Christchurch and if you have time pop over to Akaroa.
Enjoy. It’s stunning.