Undo Archive in iPad Mail After deleting or archiving an email in Mail on iOS 7, is there anyway to undo that action?
Yes, the "shake to undo" idiom on iOS works to undelete or unarchive a message. You need to perform the shake fairly quickly after realizing the mistake, since if you switch apps or go to the home screen, the undo status goes away rapidly.

Also, you can always browse the trash|archive folder and re-move messages back, assuming you can locate that message by sight or depending on the current inbox sort (date range, etc…)