I have a microphone circuit very similar to below but I don't have a mic which is on back order. I would like to test the circuit (and the processing that happens downstream) in the meantime. Is it possible to hook up the electrical signal that comes out of an MP3 player (via the audio jack) to this circuit in lieu of the mic? What other changes would I need to make?
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.
A mic might put out something like nominal -60 to -40 dBV which is roughly 1-10mV. This varies a lot depending on the actual source and microphone itself, but that should give you a rough idea of the signal levels you're dealing with.
You can use the microphone's sensitivity rating to figure out how much voltage it will produce with a given SPL in the air. But for a simple test like this that's probably overkill.
Typical mic source impedance is roughly around 150Ω or so. You can also check the spec sheet of the mic you ordered.
So for your test you could send some signal out of your mp3 player, turn it down until it reads roughly in the range of 1-10mV, and add a series resistor to bring your source impedance up to around 150Ω.
You would need to remove R1 as @EdgarBrown mentioned as that could damage your player.
Hope this helps.