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?][1]


  [1]: https://stackoverflow.com/questions/20485525/magento-custom-column-in-adminhtml-product-grid-which-contains-custom-output-l