Aoa bros

I am facing an issue in my civic exi 05 , car works fine but after driving few kilometers like 2-3kms or more the car doesn't start like battery is low, sometimes and even if we jump start or push start it works but if we turn the car off again the car doesn't start but if we wait 10-15 minutes car automatically starts without push start and jump start so i wanted to know what could be the issue? Battery is old but works fine afaik , somebody told me its related to AC and blower? as it is always turned on when the car is start . car's fan is direct