I am thinking that would work, but those are a lot of parts, just to supplant a power switch 
My thought was a hybrid of this, using what looks like 3 resistors and 2 transistors...the hold voltage comes from a dedicated uC pin, programmed to go high on startup and stay that way as needed. Drop the pin to shutdown...
No time for a schematic right now, but essentially in text...
Existing momentary switch (with pullup resistor) leg not grounded is tapped to the gate of a pmos transistor to act as momentary switch....Its source and drain attach to constant +5v and the Arduino's 5v supply input. Shorting the gate to ground conducts power to the Arduino momentarily....
A dedicated Arduino ditigal pin assigned as output would wire through a series resistor to the base or gate of another transistor, whose source and drain would parallel the other one, connecting +5v and Arduino 5v input.
Once powered up, the digital line would maintain power to the Arduino via programming. The existing switch would still do its job as a momentary grounding input switch as always, and would be unaffected by the pmos, as it is connected via the gate. No added switch, and just a couple discrete inexpensive parts....
I don't see any down side to this simplified solution. I could be wrong.