Just leaving the code outside of the <?php ?> is the equivalent of using echo(); and it becomes easier to manage. So your code could be re-written:
<?php
if( strtotime($NoticeCheck->getColumnVal("NoticeboardDate")) >= strtotime('-31 day') ) {
?>
<a href="noticeboard.php" class="responsive-picture newNotice" title="New Notice" ><picture><img alt="New Noticeboard Post" src="./images/misc/notice.png"></picture></a>
<a class="link-text newNotice" href="noticeboard.php" style="z-index:998;">XXXX</a>
<?php
}
?>
at that point it should be easy to add the other echo.