Where do I have to start? - RPM reading and action at certain engine speed

No this is a spare one. Exactly the same if the one which is mounted on the dyno

A three wire sensor typically will have a digital output irregardless of the internal device.

Now that you have the Arduino, here's what I recommend.

There is are example programs with the ide. Go to

File>examples<>02Digital>DigitalInputPullup.

Power the sensor with 5v/gnd from the Arduino. Connect the third signal wire to pin D2 of the Arduino. Run the program and bring a chunk of steel to and away from the sensor. Can you see the led go on an off or the serial output switch from 1 and 0?

If you don't see a response, the sensor may not be working at 5v. What voltage did you find on the A prong which powers the sensor?