Alternative to Port 80 for http
Solution 1:
No, you can't run on a different port than port 80 without specifying the port in the URL.
But why can't you just set your web server to provide the first site when the Host: header matches "*.example.com" and the second site when it matches "*.domain2.com"? (Someone might even be able to tell you how to do that if you tell us what web server you're using.)