Using an adapter would work but if anything ever went wrong, like a live wire inside the A/C coming in contact with the housing of the A/C then there would be an electrical shock hazard. The thing to do is to replace the outlet with a GFCI outlet (the kind with the two little buttons on them, like you should have in your bathroom). This would be a 3 prong outlet, and although the third prong still wouldn't be connected to anything, you would be protected against electric shock should something go wrong.
These GFCI outlets are not hard to install but you should know what you are doing. They come with instructions and there are videos on YouTube. Also, if the GFCI is installed as the first outlet in the chain of outlets then all of the other outlets downstream will also be GFCI protected.
These GFCI outlets cost about $15 at any hardware store or home center. And for about $7 more you can get a circuit checker that you plug into the outlet to make sure everything is hooked up right. It has 3 lights on it and they light up to indicate if everything is OK. Since you have no ground hookup (to the third prong) the tester would indicate that but of course you already know that. If you don't feel confident in doing this safely yourself then get someone else to do it. It is simple enough to install so you don't need an expensive electrician, maybe a friend or the landlord or anybody like that who has done it before. But do get the tester to check the wiring before using the GFCI.