How can I use an Arduino to simulate button presses on this Remote Control PCB
This is just the first step towards your goal. You need to figure out  whether the buttons, when pressed, connect the chip input to battery negative (pulling the input low) or battery positive (pulling the input high).







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Low-side switching and high-side switching.


Switch your multimeter to DC volts and connect the COM lead to battery negative.
Connect the V lead to one side of SW1 and note whether you get a low (0 V) or high (+12 V).
Now press SW1 and see if the voltage changes. If not then move to the other side of the switch and repeat.
Repeat the test on the other switch.


If you have the arrangement of Figure 1a the solution may be fairly easy.