How to actually make menu walker alive on admin interface?

You use the walker in your theme, wherever you call your nav menu.

So for example, if you currently have a “topnav” menu in your theme, like this:

wp_nav_menu(array('menu' => 'topnav');

and your custom walker is called “myCustomWalker”, you would update the wp_nav_menu call like this:

wp_nav_menu(
    array(
        'menu'   => 'topnav', 
        'walker' => new myCustomWalker()
    ) 
);

Just make sure to work in a child theme, so that when your theme updates, your call to the new walker (and the walker itself) are not lost.