Display success message after wp_update_post

wp_update_post

The ID of the post if the post is successfully updated in the
database. Otherwise returns 0.

So just test again that.

if ( $post_id != 0 ) { // success!
    write_here_show_success_messages();
} 

Or you can use an action.

<?php
/** use action for success message **/
if ( $post_id != 0 ) { // success!
    add_action('form_message', 'write_here_show_success_messages' );
}
?>
<div>
  <?php do_action('form_message'); ?>
  <form>
     <!-- form contents -->
  </form>
</div>