Adding 'menu order' column to custom post type admin screen

OK - in the end turned out to be fairly simple - as I'd had some kind of mental block - menu_order is a variable in the $post object (thanks to @brady for reminding me of that).

@scribu's post on creating sortable column values then gives the rest.

So, assuming the custom post type is called header_text, these are the functions and hooks that are needed:

Add a new column for the order

/**
* add order column to admin listing screen for header text
*/
function add_new_header_text_column($header_text_columns) {
  $header_text_columns['menu_order'] = "Order";
  return $header_text_columns;
}
add_action('manage_header_text_posts_columns', 'add_new_header_text_column');

Render the column values

/**
* show custom order column values
*/
function show_order_column($name){
  global $post;

  switch ($name) {
    case 'menu_order':
      $order = $post->menu_order;
      echo $order;
      break;
   default:
      break;
   }
}
add_action('manage_header_text_posts_custom_column','show_order_column');

Set the column to be sortable

/**
* make column sortable
*/
function order_column_register_sortable($columns){
  $columns['menu_order'] = 'menu_order';
  return $columns;
}
add_filter('manage_edit-header_text_sortable_columns','order_column_register_sortable');

@anu let me to the right direction, however that is not a modern code

Modern solution && working on WordPress 5.4

1. add support
2. register column
3. display column value
4. make the registered column sortable / say by what it should sort

...

$MY_POST_TYPE = "flowers"; // just for a showcase

// the basic support (menu_order is included in the page-attributes)
add_post_type_support($MY_POST_TYPE, 'page-attributes');

// add a column to the post type's admin
// basically registers the column and sets it's title
add_filter('manage_' . $MY_POST_TYPE . '_posts_columns', function ($columns) {
  $columns['menu_order'] = "Order"; //column key => title
  return $columns;
});

// display the column value
add_action( 'manage_' . $MY_POST_TYPE . '_posts_custom_column', function ($column_name, $post_id){
  if ($column_name == 'menu_order') {
     echo get_post($post_id)->menu_order;
  }
}, 10, 2); // priority, number of args - MANDATORY HERE! 

// make it sortable
$menu_order_sortable_on_screen = 'edit-' . $MY_POST_TYPE; // screen name of LIST page of posts
add_filter('manage_' . $menu_order_sortable_on_screen . '_sortable_columns', function ($columns){
  // column key => Query variable
  // menu_order is in Query by default so we can just set it
  $columns['menu_order'] = 'menu_order';
  return $columns;
});

If you are interested how would you make sortable column that is not based on some basic Query variable you would play around request filter as described in documentation here: https://make.wordpress.org/docs/plugin-developer-handbook/10-plugin-components/custom-list-table-columns/#sortable-columns

It's been too long, but just for the record, you can display the 'menu order' option in the admin, just by including 'page-attributes' in the 'supports' option array. For example:

    register_post_type( 'columna',
    array(
        'labels' => array(
            'name' => __( 'Columnas' ),
            'singular_name' => __( 'Columna' ),
        ),
        'supports' => array( 'title', 'thumbnail', 'excerpt', 'page-attributes' ),
        'public' => true,
        'has_archive' => false,
        'menu_position'=>5
    )
);

You have to register your CPT with:

'hierachical' => true