Quick and dirty solution: put the web site's name in the
hosts file of your client computer.
This way, when you type http://web.site.name/ in your web browser, it will resolve to the server's IP address and the browser will use exactly that name when making its HTTP requests.
You can do this for multiple names at once, and the browser will behave correctly (i.e. as you want him to) with all of them.
Sorry, I misunderstood the question; I thought you wanted to access one of your web sites without having an actual DNS entry pointing to it.
If you actually want to put the IP ADDRESS in your web browser and have the server answer with one specific site, you'll need to either have it run as the default web site on the server, or use another IP address and/or a custom port.