Assuming that the column combination of ItemTypeID, Service and OrderID is UNIQUE, the below code should do what you need

INSERT[dbo].[tblOrderItems]( [ItemTypeID], [OrderID], [Service] )

SELECT[ItemTypeID], @newOrderId, [Service]

FROM[dbo].[tblOrderItems] A

WHEREA.OrderID = @OrderID

INSERT#temp_result( OldOrderItemID )

SELECTA.OrderItemID

FROM[dbo].[tblOrderItems] A

WHEREA.OrderID = @OrderID

UPDATER

SETR.NewOrderItemID = NOI.OrderItemID

FROM#temp_result AS R

INNER JOIN [dbo].[tblOrderItems] AS OOI

ON R.OldOrderItemID = OI.OldOrderItemID

INNER JOIN [dbo].[tblOrderItems] AS NOI

ON OOI.ItemTypeID = NOI.ItemTypeID

AND OOI.Service = NOI.Service

AND NOI.OrderID = @newOrderId

WHERER.NewOrderItemID IS NULL