The radio has a battery on the circuit board that preserves stations when the main battery is removed.
If I may be so bold as to suggest, what you are experiencing with the A/C is a separate problem. When the air conditioning system detects low pressure, it will prevent being turned on and flash the air conditioning LED light three times.
Either you are low on refrigerant, or you have some other problem preventing the system from functioning properly. But my money is that you are low on refrigerant. If so, that means you also have a leak in the system somewhere. Take it to a shop for a proper diagnosis unless you feel comfortable tackling it yourself. If you do, lots of good people here will offer links and resources to help you do it. Including me.