Making WordPress’ page and menus visible only for logged in users, checking session variables

I think @Milo’s comment is correct. Presumably you want to use functions like current_user_can(), etc.
Alternatively, if this is just to show a different menu, you could create a custom menu for remote users then set a cookie on your wordpress site (via a call from the API) and display a different menu to your remote users when authenticated remotely.

add_filter('wp_nav_menu_args', 'wpse_wp_nav_menu_args_filter');
function wpse_wp_nav_menu_args_filter($args = array()) {
   if ($_COOKIE["remoteuser"] == "loggedin") { // User is remotely authenticated
        // Setup Primary Menu
        if ( $args['theme_location'] == 'primary' ) {
            $args['menu'] = "remote_user_menu";
        } 
        return $args;           
    }   else {
        return $args;       
            }
}

You could then use something along the same lines to determine whether a page should be visible or not.