How can I redirect users on the new 404 page without plugin?

You can assign the 404.php as a template to your custom 404 page created with the page builder.
You would need to edit the 404.php to something like this:

<?php

/**
 * Template Name: 404
 */

get_header();

$query = new WP_Query([
    'posts_per_page' => 1,
    'post_type' => 'page',
    'meta_key' => '_wp_page_template',
    'meta_value' => basename(__FILE__)
]);

if ($query-> have_posts()) {
    echo do_shortcode($query->post->post_content)
}

get_footer();

This way you would load the contents of your custom 404 page into the 404.php template.
You probably need to adapt the code above to the actual HTML structure of your theme.