If you don't remember the password for the old Apple ID, you won't be able to update the app.
If the email address still exists, use this site to reset the password of the old Apple ID:
Recover Your Apple ID - Apple
It should be possible to update the app afterwards.
Apps are connected to the Apple ID with which they were downloaded/purchased, and can't be transferred to another ID.
If you wish to use the app, and keep it up to date, but you don't remember the password, or can't change it, you'll have to remove it from your iPhone and purchase it with your current Apple ID.