here is a function that does just that:
function get_lastest_post_of_category($cat){
$args = array( 'posts_per_page' => 1, 'order'=> 'DESC', 'orderby' => 'date', 'category__in' => (array)$cat);
$post_is = get_posts( $args );
return $post_is[0]->ID;
}
Usage: say my category id is 22 then:
$last_post_ID = get_lastest_post_of_category(22);
you can also pass an array of categories to this function.