The pin 23 is the backlight 12V+ and 24 is backlight ground. I dont understand why need an extra ground for the backlight???
That circuit should work as far as I can see, but you could do just as well by arranging one Arduino pin to be continuously generating a square wave output and driving the "pump" with the two capacitors and two diodes.Since the current requirement is so low and you can "pump" at a high speed, you could use 1N914 or similar diodes and 1µF capacitors.
According to my mental arithmetic, with the components you specify, that circuit will oscillate at upwards of 50 kHz. If you are using silicon power diodes of the 1N4004 sort, they cannot switch at (anywhere near) that speed, so they capacitively load the circuit instead of rectifying. I think you might do a lot better using 1N914/ 1N4148 diodes and 1µF capacitors and use a 0.047µF capacitor for the time constant.In any given design, you may have an Arduino pin that is continuously being clocked at a relatively high frequency which you could use to drive the "pump".
Can i ask you to draw a circuit with your components? Or replace them in my circuit drawing? Sure i used 1N4001 diodes. Is it better with this diodes? http://media.digikey.com/pdf/Data%20Sheets/Rohm%20PDFs/RB050LA-40.pdf