You need to think of your problem in small pieces.
Write a short sketch that allows you to switch an LED on and off using a push button.
Write a short sketch that causes the servo to move when a variable is HIGH and move differently when the same variable is LOW.
Then it should be easy to combine them by using the push button to change the value of the variable.
Write another short sketch that lights an LED for a set duration using the technique in the Blink Without Delay example sketch. When you understand how that works you can add the concept to your servo sketch.
...R