Hi Annie,
Usually the connections are free. Water and Power connections take 1 to 3 days... Internet Connections depending of the location 7 to 15 days
Here is my recommendation:
You could start looking for possible unit/apartment/house one month or two weeks before you come here... and you can start talking to the property manager, explaining your situation and your needs
I did that when I came to Auckland, and on the second day I was visiting 3 places, and the four day I was moving to my new place
Renting market in Auckland is very aggressive... so be prepare! Units/Apartment/Houses comes and goes very quickly...
And the Best place in Auckland Central? that will depend of what you are looking... I live two blocks from my office, I can see the ocean for my balcony, and the place was brand new when I moved in.
The good news, is that there is a lot of development around Auckland, and we have seen a lot of NEW units/apartments coming to the market every three months
Sometimes, to secure a place, you could offer to pay 6 months in advance... or you could said you are interest in living there for 2 years...
Good Luck!