Check if specific user is online outside wp

You could always try using the wp_login action hook:

function custom_check_for_user($user_login, $user) {
    if($user->ID == 123) { // Where 123 is the particular user's ID
        // Do something here, eg. PHP mail() function
    }
}
add_action('wp_login', 'custom_check_for_user', 10, 2);

Further reading: https://codex.wordpress.org/Plugin_API/Action_Reference/wp_login