Get Wordpress Featured Image "alt"

Here's a solution:

  $thumbnail_id    = get_post_thumbnail_id($post->ID);
  $thumbnail_image = get_posts(array('p' => $thumbnail_id, 'post_type' => 'attachment'));

  if ($thumbnail_image && isset($thumbnail_image[0])) {
    echo '<span>'.$thumbnail_image[0]->post_excerpt.'</span>';
  }

Or you can use your code, but instead of echoing $alt directly you need to echo $alt->post_excerpt.

Check if you get correct thumbnail id.
For me this code works perfect:

$thumbnail_id = get_post_thumbnail_id( $post->ID );
$alt = get_post_meta($thumbnail_id, '_wp_attachment_image_alt', true);