1 - You don't need a programmable microcontroller or software. 2 - The regular Arduino isn't fast enough for high-quality real-time audio, the ADC is only 10-bits, and there is no DAC.3- You can't make a 180 degree phase inversion with a delay unless you have a known-fixed frequency because the period (time) of one cycle depends on frequency.You can make an inverting amplifier with an op-amp. You can also make summing amplifiers and differential (subtracting) amplifiers.Since you'll probably be adding a mic preamp, headphone amp or power amplifier that you might not be building yourself, some audio amplifiers will invert (and you won't find that in the specs because it doesn't affect sound quality) so you may not need the inverting stage. NOTE - Noise reduction headphones are fairly straightforward because the microphones are very-close to your ears and are picking-up the same noise/phase as your ears. And, the speaker/output is isolated from the microphone by the headphone housing. (It still requires quite a bit of "engineering".) Acoustic noise cancelation (without headphones) is not so easy. The intensity and phase of the noise depends on the listener's position, and you have to be careful about feedback from the microphone. (You have something very similar to a PA system with a microphone and speaker and you can get feedback-squeal if the sound from the speakers gets back into the mic.