If you have anything to add to this, please feel free to post it.
I've only had to deal with a few publisher sites, and all were done without me seeing what I was doing. Publisher is not officially supported, however, it is possible to publish with it on our servers.
When you go to publish in Publisher, there are several options. Choose the "FTP" option.
Type ftp.yourdomain.com for the location, and it should ask you at some point for your username and password, which are the same as your cPanel username and password. it should ask you what directory or folder you want to place the files in on the publish to location.
Choose "public_html" (without the quotes) and publish the site. It should work from there. Hopefully that gets you on the right track. If you have anything to add, please do so in this thread.