Even if they make a breakthrough in self-driving cars tomorrow, it could still be many years before laws are updated to enable them to legally operate self-driving cars (without drivers present) in all the different states and countries that they do business in. Insurance companies would also need to be willing to insure these cars before they'll be legal to drive.
To operate self-driving cars, they'd need to buy or finance the vehicles, which is a huge expenditure. They'd also need to hire people to clean and maintain them, since drivers wouldn't be doing that anymore. They'd have to bear the cost of fuel (whether gas or electricity) and insurance, which drivers currently pay for. Once they pay for all that, would it cost more or less than paying a driver who provides their own car? I certainly don't know.
Also, even if all the factors noted above balance in favor of self-driving cars in the long term, it may be a while before self-driving cars come down enough in price to compete with human drivers in conventional vehicles - it takes significant engineering work to evolve a design from a prototype to mass production.
The question is whether Uber can stay in business long enough to reap the benefits of self-driving cars. I don't think the answer is obvious.