Light LED when 2 others are disconnected
You want the green LED to be on only when the red and blue LEDs are both off. In other words, you want the voltage on the green LED to be LOW only when the voltages on the red and blue LEDs are both HIGH. That is a logical NAND function.

Your right-hand circuit with the RGB LED won't work because the transistor's collector is connected to the wrong place - it's shorting out the battery! Another problem is that the Arduino can only switch to Ground or +5V, so it can't replace a 'floating' switch. 

You need a configuration that produces a logical NAND function. Your left-hand circuit is doing it, but won't work with a common anode RGB LED. So swap the transistor to a PNP  type, and add a couple of resistors to switch it on when either input goes low, like this:-