If you haven't applied any of the transforms on your duplicate, pressing Alt+G should put it back where the original object was. Repeat that shortcut replacing G by R then S to cancel object rotation and scaling.
On the other hand, if you have applied transforms (with Ctrl+A), or selected all vertices then moved everything in edit mode, or moved the duplicate's origin in any way, then you will have to play around with the set origin and the cursor snap option. I would do it like this:
- In Edit mode select one vertex from your duplicate (one that hasn't changed position), then press Shift+S and click on "Cursor to Selected"
- In Object mode go to
Object > Set Origin > Origin to 3D Cursor
- Select the original object and in Edit mode select the same vertex, then like in the first step set the cursor to that vertex.
- Back in Object mode select the duplicate and do Shift+S and click on "Selection to Cursor"
... exactly overlapping faces don't get along. They usually fight over which one should be displayed and when Blender doesn't find a clear winner it ends up showing a bit of both which looks glitchy and wrong. One way around that is to scale up a tiny bit the part you want to show. Or even better, delete that part of the original mesh, then you can even merge back the two objects into one.