If I understand you question clearly that we just need to get post in that submenus from only cat id = 4 , this will make your archive.php to work only for cat id = 4
Please add below code to functions.php:
function wpse75668_filter_pre_get_posts( $query ) {
if ( $query->is_main_query() && ! is_admin() ) {
if ( is_date() ) {
$query->set( 'cat', '4' );
}
}
}
add_action( 'pre_get_posts', 'wpse75668_filter_pre_get_posts' );
Thanks!