How to bypass Drupal page cache if a certain cookie is set on a shared domain?
I would suggest looked at the Single Sign options of Drupal or implement some code that logs you into Drupal when they login to the non-Drupal site.

EDIT: first, boostrap Drupal and then run this function:

function drupal_user_login($username, $password)
{
    $uid = user_authenticate($username, $password);
    if($uid)
    {
        $form_state = array();
        $form_state['uid'] = $uid;
        user_login_submit(array(), $form_state);

        return true;
    }
    else
    {
        return false;
    }
}