I don't think you should be looking at an NPN as an inverter. That's going to mess you up in this case.
Yes, you can invert a signal with it... but you need to see what you are working with here.
The 555 will hold pin 3 LOW until triggered, then it will briefly go HIGH. So what does this do with an NPN. The NPN will not conduct (at all) until the BASE is more positive (IE 3 is high). When it conducts, it complete the path to GROUND, supplying a complete circuit for the LED and pulling the RESET PIN low on the Arduino causing it to reset. The resistor to V+ on the reset pin HOLDS it high until PULLED DOWN but the conducting transistor. When PIN3 goes LOW, then the path to GROUND goes away when the transistor stops conducting, LED goes out and Arduino no longer has a /RESET condition on the pin.
The resistor to V+ is critical to getting the right voltage levels on the pin, since when the transistor stops conducting, it is effectively removing itself from the circuit.
With a PNP, you get NONE of that nice behavior in this circuit as you have drawn. Since the PNP would want to conduct when when the base is LOW relative to the Collector. (a condition you cannot meet with this circuit) and you are always "driving" the transistor when the 555 is IDLE and you STOP satisfying the BASE when the PIN goes high.
EGAD... I'll stop here... sorry for my ranting in advance.