The project has a simple CB...
A circuit board? 
A CB radio? Usually, we write "PCB" for printed circuit board.
...with 4 leds, each with a corresponding momentary push button. I need the arduino to randomly turn one of the four LEDs on...
No problem.
... and put its corresponding switch to a LOW state (all other switches will be HIGH until its their turn in the sequence).
That's a problem. A switch is activated by the user. The Arduino cannot change the state of a switch. But of course, it can remember the last button pushed and/or take various actions depending on what switch is pushed next. Or, you can do something else (activate a buzzer) if no switch is pressed before the timeout.
And, do you want the buzzer to sound if the user pushes the wrong button? Of course, that's easy too (with an "or" statement).
...when a switch is pushed, it shorts a capacitor (on the CB), preventing its full charge. If the cap is alowed to charge, it trips a relay circuit to initiate a buzzer.
it's possible to use a resistor & capacitor as a timing circuit... I built an "analog" alarm system that way about a a milllion yoars ago... but the Arduino has a built-in clock-timer, as you probably know from the 'blink" and 'blink without delay" example programs. In your application, you need to do something similar to blink without delay, so that your program can run and watch for a button-press while the clock is '"counting-down" in the background.