Your pictures explain much more that you have written. The Diesel engine is one probably made in India. They were imported into the USA in the 1990's for use with off-grid generators and water pumps. Then disappeared.
The engine is very low compression and can be hand-cranked to start. One of the pictures shows the front shaft crank has been used quite often.
That said, the starter motor does not work very hard to start the motor. Don't monkey with the governor. To shut the engine down, put a normally open solenoid valve in the fuel line. There is no pressure in the line, only gravity feed. Close the solenoid for a time, to be determined, until the engine stops.
The engine will not come up to the rated 1500 rpm very quickly, so monitoring the generator voltage to determine how long to engage the starter is not a good idea. You need something else to tell you the engine is running. Perhaps a pressure sensor in the exhaust. Or just a piezo fastened to the exhause to detect the first engine firing.
Looks like a good project for your Arduino. Not complicated at all!