2012 Macbook Air Mt Lion 10.8.2. Can I factory reset and upgrade using only an external monitor?
Yes, it is possible, as long as the machine is run in clamshell mode.

Turn the computer on, plug in the external display, make sure an external keyboard and mouse is connected (wired or Bluetooth), then close the internal display. The Mac will enter clamshell mode and will use the external display as the primary display.

The only caveat with this is that the laptop screen must stay closed and AC power must be connected.

You can read more about clamshell mode in Apple KB article HT3131:


Mac notebooks: How to use your computer in closed clamshell (display closed) mode with an external display