WordPress Order by date on custom post type archive page

see https://codex.wordpress.org/Class_Reference/WP_Query#Order_.26_Orderby_Parameters for more information. Slightly different to what you’ve already tried, try creating a new WP Query object.

EDIT to include pagination parameters in args, try something like

$paged = ( get_query_var('page') ) ? get_query_var('page') : 1;
args = array(
    'orderby' => 'date',
    'order'   => 'DESC',
    'post_type' => 'post',
    'posts_per_page' => 5,
    'paged' => $paged
);

$query = new WP_Query( $args );

while ($query->have_posts()){
    // do stuff
}