How to fetch custom post by Author?

Question

You are using 'author' => ... twice in your code.

To get the author’s ID, you should use get_the_author_meta('ID') instead. So, remove the second author argument, and use this in your code:

  $author = get_the_author_meta('ID');
    $args = array(
        'posts_per_page' => $per_page,
        'author'=> $author,
        'post_type' => 'ultimate-auction',
        //'auction-status' => 'expired',
        'post_status' => 'publish',
        'offset' => $pagination,
        'orderby' => 'meta_value',
        'meta_key' => 'wdm_listing_ends',
        'order' => 'DESC',
    );

By the way, the global $post itself contains the ID of post’s author, which you can get it by using $post->post_author;.

If you want to use the author’s name, you can use 'author_name' => ... together with get_the_author_meta('nicename').

Take a look at author parameters of WP_Query in the WordPress codex for more details.