Can plug an air-conditioner into a outlet with an adapter
The most important thing to consider is how much current the air conditioner requires and how much current the receptacle can supply. Most USA wall outlets can supply 15 amps. Kitchen counter and bathroom outlets have long been required to supply 20 amps, but probably not at the time before grounded outlets were required. You need to first check the current requirement marked on the AC. If it has a standard 15 amp plug, it should require no more than about 80% of 15 amps, 12 amps or 13 amps. Check the fuse or circuit breaker for the outlet you want to use to determine the maximum current it can supply. Also check what other outlets are on the same fuse or breaker. After plugging in the AC, there may not be much capacity for other things plugged into the same circuit. If the circuit has the capacity, you can connect the AC.

Rather than use an adapter to accommodate the grounded plug, it would be better to install a GFI outlet in place to the old ungrounded outlet. I believe that code still allows that, but the outlet must be marked as ungrounded using the label provided with the outlet. (I checked your profile to determine that we are talking about a USA location.) Better than that but more expensive would be to add a ground wire. An electrician would need to determine how that can be done in an acceptable way.

If the AC does not have a standard grounded plug, it must require more than 13 amps. If it requires no more than 15 amps, it could be connected to the proposed circuit, but code requires that there be no other outlets on the circuit. If it requires more than 15 amps, it requires a higher capacity circuit.