OK, but is the "old format" version a different Apple account than the one you are currently using? In other words, did you set up a brand new Apple ID for the current one, or did you go into the old Apple ID and EDIT it to the new name?
If you set up a brand new ID, then that means that the apps that are asking for that ID were purchased with that ID. It is probably trying to download them as a part of the backup process, and it needs the old Apple ID password to do that.
If that is the case, and you don't remember the password, then go to: Apple - My Apple ID, and answer the Security Questions so you can change the password. I would recommend making the password to that ID the same as the password to your current ID so that regardless of which one pops up asking for a password, you would just type in the one used for both.
If you don't remember the answers to your Security Questions, you will have to contact Apple Support to get a reset link (they will want you to provide an answer to at least one of the questions): ACCOUNT SECURITY SUPPORT
Or - the other option would be to identify the apps purchased with that older ID, and to delete them and re-purchase (or re-download them if they were free) them under the current ID.
Best of luck,
GB