How do I get notified if I face any run time errors in my background running functions such as Cron or an event functions?

You can create an application Alert for your Cron functions or specific logtype written to the application logs for any code breakage, or unhandled exception or function timeout which has been occured. You can also specify the threshold for the exception count. When the result of the alert condition has crossed the threshold, all the emails which have been configured for alerts will receive an email to notify the code breakage.