Is it safe to apply voltage to the output of UA78M33CDCY regulator?
Most likely (but not guaranteed) yes, as long as there is nothing connected to board power input.

The regulator datasheet implies some voltage difference is OK:


  In the event of an input short circuit or another case where the output voltage can be higher than the input, a diode shunt can be connected across the device with the anode at the output and cathode at the input


This means that the regulator can tolerate input being 0.5v (diode drop) lower than the output. 

Can you run it without extra diode? In my experience, yes. Giving 3.3 v to regulator output raises input to 2.8 v or so. If there is nothing else on that regulator input, everything works fine. While I never had problems with this method, you can always add a reverse diode as datasheet says if you are worried.