It looks like it is depending on the onkeypress event, which isn't fired when clicking on the arrows.
Try copying the onkeypress code and adding it to the onchange event as well like:
<input type="number" id="whiskyCART_Quantity_<?php echo $whiskyCART->DisplayInfo("WAUID"); ?>" name="whiskyCART_Quantity_<?php echo $whiskyCART->DisplayInfo("WAUID"); ?>" size="3" value="<?php echo $whiskyCART->DisplayInfo("Quantity"); ?>" onkeypress="waec_showUpdate('whiskyCART_Update_Quantity_<?php echo $whiskyCART->DisplayInfo("WAUID"); ?>',this);" onchange="waec_showUpdate('whiskyCART_Update_Quantity_<?php echo $whiskyCART->DisplayInfo("WAUID"); ?>',this);" min="1" max="99" style="width: 50px;padding: 5px;font-size: 14px;" />