I have an old OM-1 in which the 1/1000 shutter speed is capping.
My google searches and APUG searches have not given me anything to work with in terms of a repair walk-through or demo.
I would like to know how to repair this myself as I'm looking to start doing more of my own camera repair and maintenance.
Repair is easy:
1. Sell / discard the OM1
2. Buy a Pentax MX
3. Enjoy a more reliable compact body with even better lenses.
Jokes aside... most likely your problem is just caused by lack of lubrication or dirtiness. So my advice is, before you even touch the curtain tension adjusters, make sure the system is well lubricated, and a bit of lighter fluid on the curtain spindles and associated mechanisms, plus lots shutter exercise might cure the issue.
In other words, do not re-adjust the curtains' tension until you are sure the mechanism has been "freed" from stickiness.
I've had many cameras and almost everytime this worked, without need of changing curtain tension.
PD: John K has gave you a good tip... take a look at the zones that would be most affected by gunk and dirtiness and start there.