I am actually from NZ
so don't let 'Japan' put you of.
One week is nowhere near long enough for many of those things you want to do. Trekking (tramping) and cycling around Fiordland are multi-day affairs and you really wouldn't do justice with any shortened versions. Milford sound is a long way from anywhere but worth the trip.
TNboy has it right in that with just one week you should do one place only and do it well. Most of the South Island is empty and you can see that scenery you want so much from the main road. With one week my advice would be to hire a car in Christchurch, cut across the Alps to Hokitika. Head south to Wanaka and Queenstown (these should be seen but don't stay unless you have big wads of cash) then further south still to Te Anau and Milford sound and then to Dunedin. So far this 4 or 5 nights.
Then a fast drive up the coast to Christchurch (a somewhat boring drive) or drive back up through Cromwell/Lindis Pass/Tekapo/Fairly (long but spectacular). This puts you back in Christchurch with 1/2 days to spare for either staying in the city or a day trip north to Kaikoura.
Of course where NZers go for holidays is a completely different story and needing at least another week and some openess from you about what you want to "see". Do you feel open punk, well do you ?
BTW Lord of the Rings was filmed everywhere so there really is no one place to go.
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