It would help quite a bit if we could see your logic circuit. Not knowing what the logic is makes it very difficuilt to design the physical part.
I remember doing the logic for this in my intro to digital design, we never got to implement it though.
A servo motor will probably be your best bet, as it will provide the necessary feedback that you need to know which position the gate is in.
I think it'll be difficult if you cant use a Microcontroller though. I havent used these before so I dont know what kind of circuits are available.
A servo motor would be nice because you can tell it to rotate say 1/2 a turn and hold that position, which would make it easier to connect the gate to.
Using the DC motor is your easiest route logic wise, you will need some limit switches that will tell you when the gate reaches the up or down position.
To tell when the person is past the gate, you could have a momentary switch mounted in the ground after the gate, that would sense the car tires going over it. once the switch is switched 2 times, you know the car has passed the gate.
Those are just some quick thoughts, I gotta get back to my homework now...