In general this may be very difficult. For example, it you have a musical recording, it's impossible to isolate a particular instrument (you can't un-mix audio). You can isolate a particular frequency (pitch*) but with real world sounds there are usually multiple overtones & harmonics, and the frequency/harmonics/overtones from different sound sources often overlap.
The Arduino may not be the best tool. A computer already has a soundcard, and if it's a laptop it's got a microphone. With an Arduino, you'd have to build the hardware and write the software (and the Arduino may not have the necessary processing power).
With a computer, you'd only have to write the software and you may not even have to do that.
If you know the audio-frequency (pitch, not the repetition-frequency) you can make a band-pass filter to block everything except the frequency of interest. And, a notch filter (or multiple notch filters) can be used to knock-out noises of a particular frequency, but real-world noise is rarely a single frequency.
A program like [u]Audacity[/u] can record the sound, and after recording you can apply filters. There iare a couple of ways to view the audio spectrum (after recording) so you can "see" audio frequencies.
Audacity won't automatically find the repetition rate, but it's a start.
I've never used MATLAB, but it (or a MATLAB clone) can do FFT or other mathematical/statistical operations on an audio file. So you wouldn't have to write an application, but you may still have to create some algorithms and do some programming-like work.
- Pitch related is the perception of frequency, A trumpet an violin playing the same note will have the same pitch and the same fundamental frequency but they sound different because of different harmonics & overtones.