1) You will be receiving information from INZ about how to pay the Migrant Levy and to get your visas put in your passports. When you know which office will be handling the passports, you can ask them directly if it will be possible for you to go there personally.
2) Use Google or another search engine. For instance, "flatshare auckland nz" brought up these hits
https://www.google.co.uk/#q=flatshare+auckland+nz, and "backpackers auckland nz" gave these.
https://www.google.co.uk/#q=backpackers+auckland+nz But there are many options - use your ingenuity to think of other keywords. And if you look at individual websites, you will see what facilities are available at the different places. If you see something that looks possible, you can ask on a thread if there is someone here who has perhaps used that place, who could give you their opinion.
3) It's possible that if you follow up an application you have already submitted, to tell the person that you now have Residence, that it may have an effect. It's worth trying. And now, all future applications can have "I have been granted NZ Residence" at the top.