It is hard to say how much they will charge you for such early check in. You really should check with the hotel itself. If it turns out that they have to charge you for another day for the room, perhaps the cheapest way will be to stay inside the restricted area of the airport for a few hours and catch a nap before heading for the hotel.
To get to Mongkok from BP Hotel in Jordan, you can walk along Austin Road to Nathan Road (just about 5 minutes) and take the MTR (subway train) from the Jordan station. There are also many double-decker buses going north from there (on the same side of Nathan Road as your hotel) that will pass by Mongkok. You can just take one of them. The hotel consierge should be able to tell you which ones. But MTR is the most convenient since you can just get off at the Mongkok station - just 2 stops from Jordan.
Hope this helps, have a great stay!