Push the left the right arrow key and servo1 rotates clockwise, release it and it stops.
Which key is "the left the right arrow key"?
You can make Processing send one value when a key is pressed, and another value when the key is released. Using arrow key is difficult, but using alphabet keys is easy. 'A' means pressed, and 'a' means released, for the A key.
does anybody know of a way to do this using just the Arduino?
Take a look at your Arduino. Do you see a keyboard? Mine doesn't have one.
You can attach switches to the Arduino, and make things happen when the switch is pressed or when it is released.