Show the category title, description and a url to it\'s archive by using it\'s category ID:

Get the Ids of the oldest and newest post in that category:

// Get oldest post ID
$posts = get_posts( array(
    \'category\' => 1,
    \'orderby\' => \'date\',
    \'order\' => \'ASC\',
    \'post_type\' => \'post\',
    \'post_status\' => \'publish\', 
) );
$oldest_post_id = $posts[0]->ID;

// Get newest post ID
$posts = get_posts( array(
    \'category\' => 1,
    \'orderby\' => \'date\',
    \'order\' => \'DESC\',
    \'post_type\' => \'post\',
    \'post_status\' => \'publish\', 
) );
$newest_post_id = $posts[0]->ID;

Sort an array of category ids by the date of the latest post:

function get_categories_by_the_date() {
    $categories_by_the_date = array();
    $categories = get_categories();

    foreach ( $categories as $category ) {
        $posts = get_posts( array(
            \'category\' => $category->term_id,
            \'orderby\' => \'date\',
            \'order\' => \'DESC\',
            \'post_type\' => \'post\',
            \'post_status\' => \'publish\', 
        ) );
        $categories_by_the_date[strtotime($posts[0]->post_date)] = $category->term_id;
    }

    krsort($categories_by_the_date);

    return array_values($categories_by_the_date);
}