Here's how order tagging in Pre-order Now works:
Customer completes checkout of an order containing a pre-order product.
Shopify notifies Pre-order Now of the new order.
Pre-order Now scans the items in the order. If the order contains a pre-order product then Pre-order Now adds the "pre-order" tag to it.
Please note that this tagging process occurs after the order is created. It can sometimes be several minutes after completion of checkout before Pre-order Now is able to add the tag to the order.