Just copy the tpl_header.php from the template_default/common directory into your custom template location and chop it to your hearts content. You'll want to move the chunk that deals with #navMainWrapper below #logoWrapper. Well thats the default anyway, I don't see many divs in the template from the site your pointing to.

As an aside, having a stylesheet_new and a stylesheet_original in the same directory can lead to head shaped dents in your desk and flying computer peripherals. I'm just saying....