Windows 7 Home Premium cannot connect to a domain. A domain is a network controlled by a server which pushes updates and settings down to the client machines when they log on.
It is unlikely that this client is running a domain, but you should ask before you jump into Home Premium, as at least Professional is required for Domain support.
Now, upgrading from Home Premium to Professional is as easy as popping your DVD in and activating the "anytime upgrade" feature from the start menu.

Shouldn't take more than an hour, but you'll have to pay the difference between Home and Pro.