How do I add Category Body Classes to Admin?

You’re using $post->ID which is undefined. Try $_GET['post'] instead:

// Get Categories
if ( is_admin() && isset($_GET['post']) ) {
    $cats = wp_get_post_categories( (int)$_GET['post'], array( 'fields' => 'all' ) );
    $cats = wp_list_pluck( $cats, 'slug' );
    foreach ( $cats as $cat ) {
        $classes .= ' category-' . $cat;
    }
}