How do I fix a sticky spot on furniture made by a rubber ball?
It depends on whether the stickiness was caused by the ball leaving a residue on the finish or the ingredients in the ball (probably solvents) ate into the finish.

If the former, you may be able to remove a residue using a wood cleaner like Murphy's Oil Soap (just one example). These cleaners are generally safe for wood finishes. If this fails, you could try stronger cleaners like Goo Gone. This type of cleaner is more likely to adversely affect finish, so try it in an inconspicuous place. However, if the milder soap doesn't clean it, you probably are risking the finish anyway.

If that doesn't work, you can try solvents, beginning with mineral spirits, but you are highly at risk of damaging finish. After wiping with the solvent, carefully wash off, again using wood soap or a mild detergent.

You can also try alcohol and lacquer thinner, either of which is highly likely to eat away the finish, but by then you are past preservation.