Xbox 360 Hard Drive exchange with Xbox 1 S

More than likely it wouldn't work correctly. It would also violate the terms of service to try.

You won't be able to play any original XBOX games with that current hard drive that you have. You will need to purchase an official 250GB hard drive or a 320GB hard drive from an official retailer (Microsoft Store, Amazon retail, Best Buy, Gamestop, etc).
If you purchase something from eBay/Amazon Marketplace re-seller and it's quite cheap and not in official packaging it's likely that it's a knock off. Since Microsoft never released 120GB hard drives with their XBOX 360 S consoles it's not possible that, that
specific console hard drive was ever official.

Official XBOX 360 250GB Hard Drive

Official XBOX 360 320GB Hard Drive

All you need is a hard drive transfer cable for $20.

Are both hard drives for the 360 S or is one for the 360 S and the other for the original xbox 360?

What do I do if my Xbox 360 won't update?

Solutions

1. Turn off your console.
2. Remove all storage devices, including any USB flash drives, external hard drives, or the internal hard drive on the Xbox 360 E console or the Xbox 360 S console. ...
   
3. Add your primary storage device; for example, the hard drive.
4. Turn on the console.
5. Try updating the console again.

@iTzZ x Burns

Yes, that is possible to do. The Xbox 360 S hard drives are compatible with any Xbox 360 S console.

If you are wondering how to remove the hard drive,
see this page for how to insert and remove a hard drive.

Let us know if you need any more assistance.