How can I test a microphone circuit with a test signal
The LM386 has quite a bit of gain which could saturate with a line-level input, but you have R2 in the circuit to compensate for that. You can set it to be a 20:1 divider so the signal level remains approximately constant.

Make sure to remove R1 from the circuit as this bias, while necessary for microphones with integrated preamplifiers, could cause distortion (or even damage) in the output stage of your MP3 player.

Keep in mind that the output of your MP3 player might be intended to drive a headphone which could have an impedance as low as 32Ω, this is unlikely to cause a problem but, if it was not properly designed, the higher impedance of your input could create distortion. You can just add a ~32Ω resistor from the input to ground if this becomes a problem.

But with those things in mind, yes, you can use your MP3 player to provide a test signal.