Airport Extreme 1st generation

The 1st generation AirPort Extreme base station came out in early 2007, and was discontinued latter that same year. As such, it would be rare that you would still have one that can still work properly.

Regardless, a "flashing green" LED, on the early models, would indicate network activity was present and being detected by the base station.

At this point, you can try performing a "factory default" reset on it in hopes that it will appear in the AirPort Utility.

To do so: Start with the base station disconnected from power. While holding in the reset button, re-apply power to the base station. Keep holding in the reset button until the status LED starts flashing rapidly, and then, let go of the button.

Ref: How to reset your AirPort base station - Apple Support

Go ahead and try a factory default reset on the AirPort Express again, but if the green rapid blinking continues, this is a sign that the AirPort Express is defective and will need to be replaced.