How to get the latest posting time of archived pages in WordPress?

Query the posts ordered by date, restricted to 1 post to display, then get the date.

$args = array(
   'posts_per_page' => 1,
   'orderby' => 'date',
   'order' => 'DESC'
);

$lastpost = get_posts($args);
echo $lastpost[0] -> post_date;

Edited last line for proper display:

$lastpostdate = $lastpost[0] -> post_date;
echo '<div class="lastpostdate>"' . $lastpostdate . '</div>';

Or, if you already have the query with the default order, you can just get the last post’s date from it:

// assuming $myquery is your query

$lastpostdate =  $myquery -> posts[0] -> post_date;

Or, if you have not altered the global query, in archives just use:

$lastpostdate = $wp_query -> posts[0] -> post_date;