Find the first block in content region that is not messages block and add class first-content-block-in-region to it.

add_css_class_to_first_block_in_region.php

<?php

function THEME_preprocess_page(&$vars) {
  foreach ($variables['page']['content']['content']['content'] as $key => &$var) {
    if (isset($var['#block']) && $key !== 'delta_blocks_messages') {
      $var['#block']->css_class = 'first-content-block-in-region';
      break;
    }
  }
}