I suspect that the simplest to program and low cost choice for controlling your servo would be one of the Arduino boards. They will have more capability than you need, but you aren't going to find anything much less expensive.
The Arduino will run off a USB and supply what you need for the pulse modulated signal to command a servo position, but I suspect that you will need a bit more juice for the actual servo power than you can get from a USB.
Many servo motor controlled arms that I have seen can be jerky and bouncy - you would want something that has some spring to it to actually touch the ipad.