Looking at the data sheets for the controller board and an Arduino Uno, you appear to have 2 options. You already have ground connected to all 3 as shown in the diagram.
- Run a wire from the 12V supply to the Arduino Vin Pin 8 Power strip
area. This is by far the easiest way.
- The L289 controller board has an on board 5V regulator
which is brought out to the blue power terminal block marked 5V Power in
the data sheet. It is specifically designated to power things like an Arduino. Use that via the Arduino's USB socket.
NOTE: There appears to be an error in either your diagram (widely reproduced in several other forums where similar questions are asked) or the L289 data sheet. Your diagram shows the 12V supply connected to the outermost connection on the controller's power connections block. This is marked as the 5V output on the L289 data sheet. This data sheet shows the 12V input to be the other end of the 3-way block with the common Power GND in the middle. You need to check this out before applying any power.
There is a lot more data out there in various Arduino specialist forums and probably the SE Arduino site. All you need to do is google for it.