Yours is a lot easier to figure out then My board you can see thee parts. lol
Just place the NPN pins collector and emitter to ground and hook the base to the arduino with a resistor 1k or higher and you should be able to make it work just as the button did.
Your button is easy if you test it one side is high and one side should be ground just solder the collector to the high side of the button and solder the emitter to the low side of the button and you'll be good to go