How to get post id of first child of the same post type?

Instead of:

    $args = array(
        'order'          => 'ASC',
        'orderby'        => 'menu_order',
        'post_type'      => 'attachment',
        'post_parent'    => $post->ID,
        'post_mime_type' => 'image',
        'numberposts'    => 1,
    );

Use:

    $args = array(
        'order'          => 'ASC',
        'orderby'        => 'menu_order',
        'post_type'      => get_post_type($post),
        'post_parent'    => $post->ID,
        'numberposts'    => 1,
    );

You could also use post/page/etc or ‘any’