Ride on something: to travel on a vehicle or animal.
I started riding on the buses.
You can ride on the horse, if you like.
Sit on someone or something: to place oneself in a sitting position on someoen or something; to hold someone or something back; to delay someone or something.
The enomous woman knocked the crook out and sat on him until the police came.
The project cannot be finished because the city council is sitting on the final approval.