internet
You won't have any problem finding an Internet terminal in the City Centre, although they are perhaps not as common or cheap here as they are in the city south. Many convenience stores have a terminal or two. Expect to pay around $2.50 per 10 min.
Customs house library at Circular Quay has cheaper access, but restricts access to email.
Paid Wi-Fi is available at many coffee shops, or adjacent to public phones. Try Clarence St for a nice, alfresco selection of cafes. For free Wi-Fi go to McDonalds, the State Library free password at the circulation desk, Customs House Library or Circular Quay Station.
3G Internet is available everywhere.
phone
Public Phones are common - you can also use them to send text messages, domestically or internationally.