Will ios 16 update recover messages deleted the day before iphone was updated?

Correct, assuming that the messages really are still in the backup.

That would mean that you backed up the phone when the messages were still on the phone.....and.....you have NOT run any subsequent backups after the time that the messages were deleted.

If you had iCloud set up to back up automatically, the very next backup after you deleted the messages would have picked up that action and deleted the messages from the backup.

Unfortunately, in most cases, the messages that a user thought were still in the backup are not longer there, so a restore produces a disappointing result.

Messages deleted with iOS version(s) prior to iOS 16 cannot be retrieved.

Axel F.

It also means that you did not sync messages to iCloud. If you did (Settings/[your name]/iCloud/Messages is OFF) then there is no way to recover them.