Electronic – arduino – Programming ATMega328 from arduino with separated power supply

How about making one of the countless serial AVR programmers and using it to write the Arduino firmware? This one worked for me, but I have USB to serial converter, so it works at amazing 3 B/s. If the programer lines on your other project are available, you can dump firmware using AVRdude. You could also probably get it form the Internet, but I don't know where to look

Yes, this is possible. See this AVR-as-RFID project for an example, which explains:

The coil actually powers the AVR through two of its I/O pins. Nearly every chip out there has clamping diodes on its I/O pins, which prevent voltages on that pin from rising above the chip’s supply voltage or sinking below ground. These diodes are useful for arresting static discharge.When you first hold the RFID tag up to a reader, the chip has no power- the supply voltage is zero. When the coil starts to pick up power from the RFID reader, these two I/O pins are presented with a sine wave, a few volts in amplitude. Anywhere that sine wave exceeds the supply voltage, some energy is diverted from the coil to the chip’s supply rails, via the clamping diode. The end result is that the chip is powered, and the coil’s sine wave is truncated. The top and bottom of the sine have been chopped off, and it looks a lot more like a square wave now.

So the clamping diodes, intended to protect your AVR from overvoltage, are conducting current from the input pin to the supply rail, powering the rest of the AVR.