I was there 11 years ago so things may be a bit different today, but as far as I know you can't go on one continuous one month walk anywhere in New Zealand. The best thing I can think of is going for a few walks in the southern part of the southern island: I recommend the Milford track (4 days), Dusky track (8-9 days), routeburn (4 days) and there are many others in that area.

Check out other possibilities in the Department of Conservation website, it seems to be as helpful online as these guys are in real life. It's a very easy country to travel in, I'm sure you'll have fun.