Hello,
I would like to ask you if it is possible use this cheap power supply:
9V battery - zener diode 5V2 - microcontroller. I have seen it somewhere on Internet. Please see the attachment. The microcontroller is something like PIC10F200 (microchip). Operating U=2.0V to 5.5V (max. 6.5)
Active mode: I=1.1mA (at 5V), sleep mode: 2uA (at 5V). Is this way healthy for the microcontroller ? Won't it be overloaded by big voltage in sleep mode ?

 

Your "regulator" is not wired correctly. The resistor goes between the input supply and the Zener.

You would be far better off omitting a regulator, and just use three AA alkaline in series to get 4.5v or four NiMH batteries in series to get 4.8v.

In order to regulate properly, the Zener you chose wound require about 20x as much current as the uC just to regulate the voltage. It would suck your 9v battery flat in about 5 hours.