Not sure I follow you totally here. What is Mysite.com? Are you your own ISP? You need to (and you probably know this - sorry), to send the message to the address of your ISP. Sounds like you may be doing this successfully with Outlook Express from your desktop? Sorry for my confusion here. I would expect your destination to be something like mail.comcast.net, or smtp.MYISP.com.
Maybe www.enteryourownwebsitenameinthisspace.com would explain better what www.mysite.com is
So Mysite is your ISP. They should be able to describe on their website what settings you need.
(1) Mail Server - the name of their mail server (firstname.lastname@example.org?)
(2) Port - you may need to specify the actual port to be used, I know I do.
(3) Username - The user name you use when you log into your Mysite account
(4) Password - The password you use
Both username and password are case sensitive.
This information should be on one of their admin pages for explanation.
Sorry, this may not help you much.