Using half wave rectifier for low power applications
I'm not sure what the third screen represents. What do you mean by "Avg sampling"?

The ripple you are seeing in screen 2 is normal. The rise is from the fast charging of the capacitor, and the fall is caused by the drain current from resistor R. I suspect that the small rapid fall in the capacitor voltage is caused by non-ideal behaviour of the capacitor (perhaps it is an electrolytic capacity with significant internal serial inductance).

To power the microprocessor, you need to install a regulator between the filter capacitor and the microcprocessor circuit. Options include linear regulators or switch-mode regulators. The former are easier. Through-hole options include LP2950-33LPRE3 or LM3940IT-3.3. In either case, you will need a second capacitor to filter the 3.3V output. Which regulator you choose depends on the expected load current, and power dissipation (power = voltage drop x load current). Your voltage drop will be rather high (about 10V ~ 13V - 3.3V), so you may want to think about changing the transformer to produce a lower output voltage on the secondary coil.