WordPress Codexによるとwp_get_attachment_metadataでメディアのキャプションが取得できるっぽい。
   Array
   (
       [width] => 2400
       [height] => 1559
       [file] => 2011/12/press_image.jpg
       [sizes] => Array
           (
               [thumbnail] => Array
                   (
                       [file] => press_image-150x150.jpg
                       [width] => 150
                       [height] => 150
                       [mime-type] => image/jpeg
                   )
               [medium] => Array
                   (
                       [file] => press_image-4-300x194.jpg
                       [width] => 300
                       [height] => 194
                       [mime-type] => image/jpeg
                   )
               [large] => Array
                   (
                       [file] => press_image-1024x665.jpg
                       [width] => 1024
                       [height] => 665
                       [mime-type] => image/jpeg
                   )
               [post-thumbnail] => Array
                   (
                       [file] => press_image-624x405.jpg
                       [width] => 624
                       [height] => 405
                       [mime-type] => image/jpeg
                   )
           )
       [image_meta] => Array
           (
               [aperture] => 5
               [credit] => 
               [camera] => Canon EOS-1Ds Mark III
               [caption] => 
               [created_timestamp] => 1323190643
               [copyright] => 
               [focal_length] => 35
               [iso] => 800
               [shutter_speed] => 0.016666666666667
               [title] => 
           )
   )
WordPress Codexより抜粋
 
$metadata = wp_get_attachment_metadata( $attachment_id );
$caption = $metadata['image_meta']['caption'] ? esc_html( $metadata['image_meta']['caption'] ) : '';
echo $caption;
これでいけるはずなんだけど、なぜか取得できず。 var_dumpしてみるもは空。これはバグなのでしょうか? どうやらget_postで取得するしかなさそうです。
$caption = ( $attachment = get_post( $post_thumbnail_id ) ) ? esc_html( $attachment->post_excerpt ) : '';
echo $caption;
参考:http://www.webopixel.net/wordpress/779.html