Would replacing an old 15 amp breaker stop the air conditioner and fan from throwing the breaker?

What size is the AC? Look at the power consumption rating of the AC unit; if it pulls anything close to 15 amps, a new breaker will not solve your problem.
There should be an information panel that shows the current draw of the AC. If the AC pulls more than the breaker can handle, you need to re-wire the outlet for the AC. You can't safely just replace the breaker with a larger one since that would result in a fire hazard.