I would say that the role of a governor does prepare a person to become president. This is one reason why so many recent presidents (Carter, Reagan, Clinton, Bush, Jr) have been former governors.
The thing about being a governor is that it involves many of the same things that being president does. This is because both governors and presidents are heads of executive branches. Governors have to set a budget and deal with the legislature, for example, just as the president does. They have to manage the executive branch of their state just as the president manages the federal executive branch.
So governors at least have some experience with managing an executive branch and dealing with a legislature. These are important things that a president must do well.