Since last_order_date
is a real field you will need to implement filter_condition_callback
$this->addColumn('last_order_date',
array(
....
'filter_condition_callback' => array($this, '_filterLastOrderDate'),
));
See Magento - custom column in AdminHtml product grid which contains custom output/logic?Magento - custom column in AdminHtml product grid which contains custom output/logic?