Window Type Aircon Upside down
Whenever I've purchased items that use refrigerant the manual always recommends leaving it to rest. Times have varied from 1 to 24 hours. The manual says this is to let the refrigerant settle after transit. 

There is oil mixed in with the refrigerant that lubricates the compressor. If it's tipped over or turned upside down the refrigerant can move to other parts of the system. If you turn on the air conditioner in this condition it may not work or may even damage the unit. Set it in the correct position and wait 24 hours before using it. Realistically less than an hour should be fine but most manuals recommend 24 hours to be safe.

So no. Moving the air conditioner upside down won't damage it. If you try to run it too soon after it's been upside down that can damage it.