Custom loop of a single category, cannot order by date. What am I missing?

If it doesn’t work, chances are you do not have at least one post linked to that specific category 4.

If you wanted to sort by DESC and by date, you wouldn’t have to specify those as arguments in the query because those are both the defaults for sorting.

Also, instead of initializing a WP_Query object and simply using the query method, try to get in the habit of passing an array of arguments to the constructor.

So in your case, try the following:

$args = array('cat' => 4, 'posts_per_page' => 1);
$announcements = new WP_Query($args);

if ($announcements->have_posts() ) : while( $announcements->have_posts() ) : $announcements->the_post(); ?>    
    <h1><?php the_title() ?></h1>
    <div class="post-content"><?php the_content() ?></div>
<?php endwhile; endif; ?>

Note, I did two things:

1. Created an $args array and passed it to the WP_Query constructor.
2. Check whether or not it returned any posts by doing if ($announcements->have_posts()) before the while loop.

Take a look at this for lots of information on the WP_Query class http://wp.smashingmagazine.com/2013/01/14/using-wp_query-wordpress/