The most common method of doing what you want is to use a lobbed cam that actuates a microswitch at the desired stop point.
To do the sequence that you want, you need to add a tiny bit of smarts. This can be a couple of relays or a microcontroller.
As mentioned in the comments, most automobiles have windshield wiper motors that have such a switch mechanism built in. However, because you want the motor to operate only one full cycle upon power-up, you do need to add the smarts that I mentioned.
