Is it permissible, in Sharia, to lease a passenger car to a driver for a fixed daily amount, with the driver being responsible for all car expenses, and is this contract considered Mudarabah?

Renting a car is permissible. For the contract to be valid, the usufruct (benefit) and the rent must be known. If the intended expenses are fuel and related costs, then it is permissible. However, if it includes repairing damage to the car that was not caused by the renter's transgression, then the contract is invalid. The contract is valid even if the duration is not specified, but it becomes binding only upon the commencement of the day (of rental). Either party may terminate the contract before the commencement of the day.