There are a few things wrong with your circuit.
First, you need an op-amp that will function on 5V. Half of an LM358 might work for you.
Second, you should divide the output voltage of the op-amp, not the power supply.
Third, you must bias the op-amp so that the DC value of the output is near the middle of your ADC range.
Fourth, your gain of 1,001 may be too large, and is certainly too large for a DC gain, you need to separate AC and DC feedback so that the DC gain is much less (close to one is good). Putting a capacitor in series with R2 would allow the DC gain to be 1.
Fifth, whatever op-amp you use, you must have a DC path from both inputs to a bias voltage. Connecting the non-inverting input to the capacitor means it will float to some undesirable voltage in milliseconds. A resistor to an appropriate bias voltage will do.
The eletret microphone requires a dc bias to enable it to function. However, you only need to detect the audio (ac) signal to amplify and feed to a speaker. The capacitor blocks the dc bias, whilst passing the audio signal to the amplifier. There must be an amplifier to generate the required power to drive the speaker. The speaker will have a low impedance, typically 8 to 16 ohms. The amplifier must provide the power and a matching output impedance _ else won't work!
Best Answer
For sure.
The electret will pick up sound just like any other microphone. There are a few draw backs though. You will be limited by the range of frequencies you can pick up with this type of microphone. I'm not sure what range you can expect but such information would be listed in the microphones datasheet.
Once you are able to get the analog stream from the microphone, you can analyze it using a suitable FFT algorithm, such as FFTW. It is from those, you'll be able to extract various peak frequencies within the music.