WordPress: category page not for post’s

Add the ‘post_type’ parameter in your query args ('post_type' => 'game'):

<?php
if (is_page() ) {
$category = get_post_meta($posts[0]->ID, 'category', true);
}
if ($category) {
  $cat = get_cat_ID($category);
  $paged = (get_query_var('paged')) ? get_query_var('paged') : 1;
  $post_per_page = 4; // -1 shows all posts
  $do_not_show_stickies = 1; // 0 to show stickies
  $args=array(
    'post_type' => 'game',
    'category__in' => array($cat),
    'orderby' => 'date',
    'order' => 'DESC',
    'paged' => $paged,
    'posts_per_page' => $post_per_page,
    'caller_get_posts' => $do_not_show_stickies
  );
  $temp = $wp_query;  // assign orginal query to temp variable for later use   
  $wp_query = null;
  $wp_query = new WP_Query($args); 
  if( have_posts() ) : 
  while ($wp_query->have_posts()) : $wp_query->the_post(); ?>
     <div <?php post_class() ?> id="post-<?php the_ID(); ?>">
        <h2><a href="https://wordpress.stackexchange.com/questions/3001/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>
        <small><?php the_time('F jS, Y') ?> <!-- by <?php the_author() ?> --></small>
        <div class="entry">
          <?php the_content('Read the rest of this entry »'); ?>
        </div>
        <p class="postmetadata"><?php the_tags('Tags: ', ', ', '<br />'); ?> Posted in <?php the_category(', ') ?> | <?php edit_post_link('Edit', '', ' | '); ?>  <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?></p>
      </div>
    <?php endwhile; ?>
    <div class="navigation">
      <div class="alignleft"><?php next_posts_link('« Older Entries') ?></div>
      <div class="alignright"><?php previous_posts_link('Newer Entries »') ?></div>
    </div>
  <?php else : ?>

  <h2 class="center">Not Found</h2>
  <p class="center">Sorry, but you are looking for something that isn't here.</p>
  <?php get_search_form(); ?>

 <?php endif; 

 $wp_query = $temp;  //reset back to original query

}  // if ($category)
?>

Reference: http://codex.wordpress.org/Function_Reference/query_posts#Post_.26_Page_Parameters