NB: Please make a backup of your page template before trying this:
This is something I use in my template to show my last 4 posts from the work category (displaying only an image that links to the posts) and have modified to show the last one post which I hope will also work for you:
<?php
//get posts from category named work
$categories_to_include = get_cat_ID('work');
$the_query = new WP_Query('cat='.$categories_to_include.'&showposts=1&orderby=post_date');
?>
<?php while ($the_query->have_posts()) : $the_query->the_post(); ?>
<div class="postWrapper" id="post-<?php the_ID(); ?>">
<div class="post"><a href="<?php the_permalink() ?>" rel="bookmark"><?php $values = get_post_custom_values("thumbnail-name"); echo $values[0]; ?></a></div>
<p class="postMeta"><?php edit_post_link(__('Edit'), ''); ?></p>
</div></div>
<?php endwhile; ?>
change category from work to name of your category
change thumbnail-name to your custom field name
Hope that helps get you on right track…