Yo might be able to do it, but I think you're "nuts" to attempt that with Arduino. 
The Arduino is "under-powered" for audio processing and there is no built-in DAC on a standard Arduino. Plus, you'd need a preamp on the input and a power amp (or powered speakers) on the output.
The [u]TMRpcm library[/u] uses "fast PWM" to approximate analog-out without a DAC but that can be "tricky". TMRpcm also has a recording feature but a lot of people have trouble with that too.
If you can record and play-back reversing the audio is easy... You just reverse the samples or read the audio file/data backwards.
Is there a reason you can't use a computer or smart phone? A computer has a soundcard and plenty of processing power and if you have a laptop it's got a built-in microphone. If you need something stand-alone, maybe check-out the Raspberry Pi. It's got some limitations too but it should be more suitable than the Arduino.