After installing Active Directory on our new server (Windows Server 2008 R2) following these requirements we lost internet connectivity on the server itself.
How can we restore it? Using the second NIC available?
This sounds very like a DNS issue, or possibly a routing one - can you ping remote hosts by IP? 18.104.22.168 (www.serverfault.com), for example?
As the linked article mentions, the new DC will be its own primary DNS server, so it will be able to see all the resource records that AD sticks into DNS. Unless you've got a forwarder configured to handle requests for things that aren't in your local DNS, you will indeed lose Internet connectivity.