we will arrive before hotel check in time, is there a luggage storage facility at the Forbidden city

Yes, there is lugguage facility in the Forbidden City nearby the ticket office. The problem is that now, visitors are only allowed to enter from the south entrance and then exit from the north Gate (Shenwu Gate). If you store the lugguage at the entrance in the south, you need to go back from the north gate after visiting. You will arrive before hotel check-in time, but you can leave the lugguge in the hotel and go out to visit. Then check in when you finish sighseeing.