Mirroring would require that you have an exact replica of the website. This replica would need to be in sync with the main website. So mirroring actually involves rsync-ing (which is actually the most efficient way). You then run a webserver to make your mirror accessible. A mirror is usually published, somehow,  from the master site, either by direct url pointing. I think lately they can use GeoIP to accurately determine your location and redirect one to the nearest mirror.

As concerns DNS, mirrorname.your.dom.ain is controlled by your DNS servers.... so your DNS admin does that. The person running the master website can determine the name as well, based on their.dom.ain and as such they can control the DNS while you handle the Apache internals (rewrite rules, etc) on the mirror. That's the much I can remember from a previous experience when  I did http://lix.in/-9bdfdc.