There is bus from Hong Kong Heritage Museum going to ChungKing Mansion?

Yes, take bus no.81C at Che Kung Miu, get to Middle Road Tsim Sha Tsui. The duration is about 50 mins. Then walk to northeast around 110 yards (100 meters) to find the destination.
The ticket fare is 7.8 HKD per person. If you using the Octopus Card or elder card, I cannot calculate the exact price. But you can actually enjoy some discounts. You can consult the staff when you arriving at the MTR Stations.