How can I enter a app, that has to be updated, purchased in a different apple ID without having to delete it?
If I understand your question correctly, you could sign out of the App Store on your iPhone and then sign back in using the other Apple ID and password. Once signed in, you should be able to update the app in question.

Note that other apps not installed under this Apple ID will now not be updatable until you sign out again and log back in with the original Apple ID and password.

The only way around this is to delete your apps and re-download them using the other Apple ID. However, it's likely that if you have multiple apps installed using each of these Apple IDs that you may face similar issues to the one you're now facing with the music app, so you may be better off just signing in and out of your various Apple IDs if and when you need to.