How to show only the most recent post on my custom post type archive?

You could just alter the main query for your custom post type archive with pre_get_posts().

Code:

function wpse124228_alter_ppp_order_for_mycpt( $query ) {
    if ( ! $query->is_main_query() || is_admin() )
        return;
    if ( is_post_type_archive( 'mycpt' ) ) {
        //Only display 1 post on mycpt archive
        $query->set( 'posts_per_page', 1 );
        //Most recent/current
        $query->set( 'orderby', 'date' );
        $query->set( 'order', 'DESC' );
    }
}
add_action( 'pre_get_posts', 'wpse124228_alter_ppp_order_for_mycpt' );

This pretty much resembles this example from the codex page. BTW don’t use query_posts(), if you have to do a second query go with WP_Query, but for an archive that shouldn’t be necessary.

Additional Information:

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