How can you update an app if you don’t know your billing password

Your Apple ID password? If not that, what do you mean by billing password?

If you forgot the Apple ID password, enter the Apple (iCloud) ID, and click Forgot Password. See if you can get a Rescue or Notification email option if necessary. Link to contact Apple Support at the end of the 2nd article.

Apple ID - Manage            

Apple ID – Forgot Password