My bathroom sink has water level going up in it, not draining. The water is cold, could it be the drain for the air conditioning system upstairs and what is best thing to do

You have blockage in the drain line you need to use a snake, thanks.

Joh

My bathroom sink has water level going up in it, not draining. The water is cold, could it be the drain for the air conditioning system upstairs and what is best thing to do

John nailed it! Let me take it farther. You have a blockage in the horizontal line in the wall the must be snaked and cleared.
Remove the "J" bend from the trap and send a snake up into the wall, (see image). You will run into a 90 about 6 inches in but once past that you only have to put out 5 more feet of cable. After, flush with a bowl of hot water.
Good luck, Tom