Sometimes the theory of the origin of a phrase is more interesting that the actual origin. I personally think that is true in the case of today's idiom. 'On the ball' is the short version of the common phrase 'keep your eye on the ball'. One of the ideas about this phrase has to do with a time-ball at the Royal Observatory in Greenwich. Read about it here and about the actual origin of the phrase. We use the idiom today in much the same way it was meant when it came into being - be alert.

The origin of this phrase is literal. When the ball is in your court, or your side of the court, then it is clearly your turn to hit it, etc. ( source )