Hello!
I want to use INA226 module to measure own supply current and voltage of Arduino using the same Arduino. Just like that:
Is it correct circuit?
Thank you in advance 
Looking at the blocks and making several assumptions I think it might work. Remember the Arduino does not draw much current.
You will need to connect the INA226 GND to Arduino GND.
Yep, of course
Does it works? Meaning, does sharing GND/VCC with measured object (Arduino) and measuring one (INA266) will be OK?
The INA226 module and Arduino require a common reference (GND) for I2C to work properly.
It should work just fine.
You didn't say which Arduino you're using.
Make sure VCC of the INA226 is connected to VCC of the Arduino.
That could be 5volt or 3.3volt, dapending on the processor's logic voltage.
Leo..
That depends on the Arduino and INA219 module you are using.
Is it good idea? Current will NOT be measured, when connected Arduino to VCC, shunt resistor in INA226 will be on VCC both sides.
@Wawa let's say IN226 and Arduino has 5V supply.
An Arduino can have a 5volt supply and still be a 3.3volt processor.
An example is a Due.
VCC of the INA is it's logic voltage, which must match VCC of the processor.
IN+ and IN- or the screw terminals of the INA are the shunt connections, where the supply current runs through.
Just tell us which Arduino you're using.
Leo..
Just Uno
Uno R3 I assume.
VCC of the INA must be connected to the 5volt pin of the Uno.
But what current do you want to measure.
The Uno itself or attached sensors or loads.
Where does the power come from. USB or Vin.
Your diagram is incomplete.
Leo..