All your circuit should work.
But they don't and I don't know why.
Nobody else knows why either... We don't know if the problem is in your microphone, amplifier circuit, or in your firmware/sketch, or somewhere else... In order to troubleshoot your project, you need to test the individual sections and "zoom-in" on the problem.
What's happening? Anything?
Do you have a multimeter to measure the signal out of your microphone amplifier? Maybe try putting +5V on the Arduino input to make sure the Arduino and the firmware are working. (With 5V, I assume the VU meter will max-out.)
Or, maybe add a [u]protection circuit[/u] (two diodes and a resistor) and feed-in an audio signal from somewhere else (bypassing the microphone) .
I found a TDA2822M and the datasheet says its some kind of amplifier. Is it useful for this project?
Maybe... But it has fixed gain, and we don't know how much gain you need. The required gain depends on the sensitivity of the particular mic, and the loudness of the sound. Usually, it's best to have a gain/sensitivity control. And, you'd still need to bias the electret mic.
Can I use a microphone like this without any additional circuit?
That looks like it should work.
I dont have one, so I'd rather make one out of parts I already have.
I'd say your odds are better with something pre-built. ;)