Microphone with LM386
Is the detection too slow? Do I have an error in my schematic?


Using Arduino as a DC voltmeter here should work since all the DC voltages are within the range of 0V to +5V. Ensure that the path to the Arduino input is DC-coupled (no intervening series capacitors).

Since Arduino's analog-to-digital converter is a sampling type, it could be too fast. When troubleshooting the LM386 circuit, no audio input signal should be applied, so that all the voltages measured are DC voltages. No-signal-input should ensure that Arduino's voltage measurements don't fluctuate.
Arduino's analog-to-digital converter may result in false voltage measurements if the LM386 circuit is oscillating - always a possibility when probing a new build.

OP's schematic seems correct.


  I also tried to replace the MIC with a DC-source and tried producing
  up to +5V.


Data sheet absolute maximum specification says that no input below -0.4V nor above +0.4V should be applied to pin 2 or 3. You may have exceeded this spec.

Before attempting to listen for audio, DC bias voltages should be checked with no MIC input signal. Using Arduino ADC as a voltmeter is a novel way to check voltages. If you have an oscilloscope - that's even better, because not only DC voltages can be measured, but any possible oscillations can be seen as well.

LM386 DC voltages, with +5V applied to pin 6:


pin 1 (internal) about +1.2V above gnd
pin 2 (input) +0.0125V or less
pin 3 (input ) +0.0125V or less
pin 4 (ground) 0.0V
pin 5 (output) close to +2.5V
pin 6 (Vcc) +5.0V
pin 7 (bias, internal) about +3.1V
pin 8 (internal) about +1.2V above gnd.