Make homepage the newest post of a custom post type

Create a front-page.php file in the theme and query for the latest post of your post type.

$args = array(
  'post_type' => 'yourtypename',
  'posts_per_page' => 1,
  'ignore_sticky_posts' => true,
  'post_status' => 'publish',
}
$latest = new WP_Query($args);
if ($latest->have_posts()) {
  while ($latest->have_posts()) {
    $latest->the_post();
    the_title();
    // etc
  }
}

See the Template Hierarchy for information about files that are “automagically” loaded..