"You can't hit a ball with THE bat of your eye" is wrong. It should be:

You can't hit a ball with A bat of your eye.

To bat one's eyes is to lower the eyelids flirtatiously.
============
I think the mystery line is "You can't bring water for punching a bag." It doesn't make much sense, but "punching the bag" either refers to boxing practice where fighters punch a hanging bag at the gym, or to going to sleep, as Uncle Wiggley says.

I believe boxing was much more on people's minds in the 1920's than it is today.