It is a misconception that Columbus set out to prove the earth was spherical as this was this was common knowledge to educated westerners since the time of Aristotle. Did have any predictions about how long it would take him to reach Japan?
No, he had severely underestimated the size, and expected to reach Asia in a small fraction of the time it would have taken has the America's not been blocking his way. Check this response for some more info on this matter!