a lm386 is nothing but an opamp capable of reasonably large current. so yes, it can be used here.
a typical electret mic outputs about 100mv. so to output to full 3.3v mcu or 5v mcu, you need about 40x gain, fairly easy with a lm386.
the original design got a lot of things right: 20x gain (jumper-able), input impedance matching to the mic, input dc blocking (I would upsize the .1u a little bit, to maybe 4.7u but since you aren't looking for sound quality, it is OK to use .1u cap here), supply decoupling, and that 1k resistor on the output.
in short, it should work.
what you want to do next depends on what it is not working for you - "circuits don't work very well." isn't exactly helpful to people who are trying to help you. exactly what are the problems you are having with the circuit?