[Captain_hook]

If you use 2 small air cylinders and connect their rods together you can use your valve to drive one cylinder and that will drive the second cylinder. On the second cylinder you the fit two check valves, to allow the water to be draw in then spat out.

 

[Captain_hook]

You could just install a stop on your cylinder set up so the 12 cylinder only gets drawn out a portion of the stroke. That way you can play around with the position of the stop until you get the amount of water you want.

You also wouldn't need to change both cylinders on the one connected to air. The length of the "pump" stroke is set by the length of the "power cylinder".
You also don't have to have a bucket you could connect the input of the pump cylinder directly to your garden hose.

Good luck.