How to detect if a light bulb is burned out or blown in Arduino

Runaway Pancake has the right approach. Though the OP will never get there with the drawing he has. If you know the approximate CURRENT of the lamp, put a small resistor in series with the emitter of the transistor, then connect the emitter to an analog input. Size the resistor such that you have a few hundred millivolts going to the input. This is how it's done in automobiles. My BMW X3 got very upset when I changed turn signal bulbs to LEDs. It was looking for more current, or a higher voltage at the analog input. Since most light bulbs are running with one end grounded, it takes a bit more circuitry to sense at the "high end" but the principle is the same.