Are you talking about an arcade game or a console controller?
Either way, you should get a voltmeter/multimeter and check the voltage difference between buttons. I'd guess it'd be 5V but maybe it's something different. If it is 5V you could potentially run the buttons from the arduino's digital pins. If not, you could use transistors as switches: the emitter and collector would bridge the two terminals of a button, and the arduino's digital pin would connect (thru a current-limiting resistor) to the base of the transistor. If you can find/afford an appropriate MOSFET that would work too.
Most buttons connect 2 terminals to let current flow. With a voltmeter, figure out what the voltage difference is across each button terminal, and which terminal is positive and negative. With an ohmmeter, find out if any of the button terminals are directly connected (they'll have zero or very tiny resistance between them). For example, maybe the "left" side of each button shares a common connection to 5V or to ground: this could simplify your project.
If you don't have any experience with transistors, check out
https://learn.adafruit.com/adafruit-arduino-lesson-13-dc-motors/transistors - the example is for a motor, but instead of switching a motor on and off you would just be making a connection between the button terminals.
It might also help to review how buttons work with digital devices. This example shows how an arduino uses buttons, a video game controller will work in a similar way because there's a microcontroller in there.
http://arduino.cc/en/Tutorial/Button