Arduino Nano: This schematic would work?
Yes. That should work as long as your Servo motor is rated 6V power (each battery cell contributes 1.5V). The key thing is that the battery pack GNDs be connected together. The servo motor responds to a variable pulse width signal. The Nano will output 5V digital I/O. The servo should be responsive to PWM at those levels, but as always check it's datasheet for maximum limits (or provide a link and we can help).