@yaantey
I happen to have a few of those mics, so I did this experiment using my scope I have. Well I set my voltage setting at 2 mV/div and I did get something, ( I was blowing on it ) and I place close to the speaker of my DIY AM radio, I did get something ( a sound waveform ) and try to measure it, it between 2 mV to 5 mV ( peak-to-peak ). If you feel the op-amp circuit do not amplify enough, you can still add another amp. Usualy, when I design this type of system, I add a transistor for the fisrt stage and the second stage a op-amp. Then I add a filter ( in your case a band pass filter set around the baby cry frequency ) and amplyfy and buffer to going into a analog input of the arduino.
And I will use a electronic circuit simulator to check the design and change some values ( resistors and capacitors ) if needed - if it improve or not.