Starting something upon button press and running it for three minutes is of course dead easy:
loop() {
if (digitalRead(BUTTONPIN, LOW)) {
startServo();
delay(3 * 60 * 1000);
stopServo();
}
}
That'll do the job. Just fill in the proper functions to start and stop the servo, define the pin connection, set the pinMode()s and so, and you're in business. Now while this solution does the job, it's also plain ugly as it blocks the processor from doing anything else while it sits out the 3-minute delay().
For wiring, ask our good friend Mr Google. You'll easily find wiring diagrams for buttons, servos, and just about anything else.