Is a simple adapter good enough to plug a 240v plug Aircon in a normal outlet?
No it is not enough. 240V outlets in the US are doubled up conductors, i.e. there are 2 legs of 120V each. The outlets have 4 holes, two for hot, one for neutral, one for ground. Eve if you had an adapter that converted the 3 hole to 4 holes (i doubt such a thing exists) it will not work because you will only pull 120V from that point.
The adapter is meant for travel to Europe where the single pole outlets are 240V.
You might be able to convert a 120V outlet into 240V if you're handy. That's my rec