Having had
labrum repair, and rotator cuff repair surgery last year, I can tell you that no, no you can't run.
I'm in good physical shape, and I had my shoulder repaired by one of the best doctors in the country for that particular type of surgery, and I was still immobilized in a sling for 8 weeks barring showering, and PT. Even after the 8 weeks I still had another 6 weeks of PT before I was cleared to ease back into light activity and weight bearing exercises, and was't completely cleared for a full 6-months after the surgery.
Trust me, just trying to stabilize your shoulder when you cough or sneeze after that sort of repair is not only excruciatingly painful, but can also tear loose the bindings used in the repair if it occurs too soon after surgery.