Hi.
OK, your diagram is a single 555 in astable mode. The reset is pulled to 5V so it can run all the time. What you need now is another 555 in monostable mode. This other 555 is set to do a 3 second time out when you press the switch to kick its input. This 3 second 555 is used to hold your astable 555 in reset when it's not triggered so the servo doesn't run. Press a switch, it triggers the first 555, which releases the reset on the servo 555 so as to run the servo 3 seconds. At the end of the 3 seconds, the first 555 goes back to its dormant state, which puts the second 555 into reset, and the servo stops running.
This stuff is all over the web, Google is your friend. Also, there is an IC with two 555s in it, called the 556. You might even find the diagram in a data sheet to do just what I described.
Have fun with your project.
kenjj