In order to get anything stored to Session, you have to submit it to the server.
Perhaps some more details on why you don't want the UpdatePanel refreshing would be helpful, and what you are trying to accomplish using the value in Session.
EDIT: Based on your comments, it seems to me that the solution would be to store the current .ascx file in Session, and set your DropDownList to have autopostback enabled.
So, on your handling of the "Next" and "Back" buttons, store an indicator for the correct .ascx to Session.
During your postback handling of the dropdownlist event, you could simply ensure that the current .ascx file is still being shown, by checking session for the correct file to show. When the result is returned to the client, nothing will appear to have changed, because the UpdatePanel is smart enough to realize it's the same content, and you will have successfully dealt with the dropdownlist value.