Display random post on a page with post permalink in URL

What you are asking is how to redirect the visitor to a random post. Here you go:

<?php
/*
 * Template Name: Random Redirect
 */

query_posts(
    array(
        'showposts' => 1,
        'orderby' => 'rand',
    )
);

if (have_posts()) : while (have_posts()) : the_post();

header( 'Location: ' . get_the_permalink() , false , 303 );

The syntax is

void header ( string $string [, bool $replace = true [, int $http_response_code ]] )

… and the 303 status code denominates «See other».