Hi @overachiever
Can you debug the $post_object value?
$post_object = get_field('post_object');
echo '<pre>';
print_r( $post_object );
echo '</pre>';
die;
Next, Try the code without using the setup_postdata function:
<?php
$post_object = get_field('post_object');
if( $post_object ): ?>
<div>
<h3><?php echo get_the_title( $post_object->ID ); ?></h3>
<span>Post Object Custom Field: <?php the_field('adlinks', $post_object->ID ); ?></span>
</div>
<?php endif; ?>