Even rounding all the numbers up, the motor is only going to need a fraction of an amp-hour per session based on those numbers. If you keep the UNO itself powered up throughout the whole four hour session, that would use roughly an additional 200 milliamp-hours. A set of ten NiMH AA cells in series giving 12V and 2AH capacity ought to be ample for that application. You could reduce the number of cells if you feel that the stepper may not need the full 12V, but that's the recommended voltage so I would suggest sticking with that. Use bigger capacity cells if you want it to be able to last multiple sessions without recharging.