OST does not do this by default.
As far as getting your server to print a copy directly there is a way to add this functionality, but it would require some coding on your part. Here is a link to the functions that would most interest you:
http://docs.php.net/manual/en/ref.printer.php(http://docs.php.net/manual/en/ref.printer.php)
I imagine that it may vary greatly depending on what OS your server is running.
The easiest way is to send your alerts to an email loop (or distribution list) and have one of the recipients on the list be a generic account that no one really has access to. Then you could configure another machines outlook to do what you are doing currently.
I'm not really a hippie conservationist, but why do you need to print out every ticket like that? Please consider saving a tree. :)