harvested a joystick from a fight con, want to use it to take input into an arduino. how do I wire up the switches?
The 16A 250VAC rating is the maximum for each switch. You can certainly use 5VDC for the Arduino.

It looks like from the pictures that the joystick simply presses one or more limit switches when it's moved. It also looks like all of the limit switches have both terminals exposed. When the limit switch is pressed, it simply connects the two terminals (or breaks the connection if it's normally closed). 

What I would do is connect one terminal from each switch to 5V, and connect the other terminals to 4 Arduino pins. Put pull-down resistors on the four wires going to the Arduino. When a limit switch is pressed, you will read a 1 at the corresponding Arduino pin, and when a switch is not pressed, you will read a 0 at the corresponding Arduino pin.

You also might want to implement hardware or software debouncing on the inputs.

EDIT:

The other method you described involves attaching one terminal from each switch to ground, and connecting the other 4 terminals to 4 Arduino pins. There must be a pull-up resistor to 5V on each input wire. Then, then the switch isn't pressed, you will read 5V through the pull up resistor, and the when the switch is pressed, you will read 0V since the pin is shorted to ground.