As everyone has said, it's very doable. However, I think you're complicating the hell out of it. Make it simpler on yourself!
A few things immediately come to mind:
A solenoid is easy enough to buy and operate, but they are often bulky and expensive. There are already digitally controlled hose manifolds for watering your lawn. Buy one of them for $20, open it up, and see what happens on the board when it triggers. Solder in wires and supply the voltages yourself!
Don't make a water column rise when the player hits the target. That's a much more involved process than you think it is; as the height rises, the pressure column builds and you'll need more force to raise it higher, resulting in a diminishing rate of rise. You can't just solenoid valve it, as then it will seek to equalize pressure and it won't raise beyond what's supplied in the pressure head. You'd either have to use a solenoid pump (which are a PITA to control) or a peristaltic pump (which are expensive, noisy, and aren't designed to move much water). My choice would be to have the target close a switch (a simple feather switch would work great). When that's closed, it completes a circuit to turn a small DC motor. The motor spins a long threaded rod ("Screw Drive") that you buy at Home Depot for $2. You have a little indicator ride a rail with a nut on the back... As the motor spins, the indicator rises along the screw! Have another feather switch at the top -- first one to hit the switch wins! (and it sets off lights/sounds/closes water gun solenoid/etc)
The electronics and coding aren't that hard. The mechanical aspect may be your toughest hurdle.