It's possible that running in the opposite direction of the rotation of an artificial gravity ring could cause you to feel weightless, but it's also possible that it wouldn't have any effect at all.

The key factor in determining whether or not you would float is the magnitude of the rotational velocity of the ring and how closely it approximates the force of gravity at the surface of the ring.

In order for an artificial gravity ring to create a sensation of weight, it would need to rotate at a high enough speed to produce a centrifugal force that is similar in magnitude to the force of gravity at the surface of the ring. If the rotational velocity of the ring is too low, then you would not feel any sensation of weight, even if you were running around the ring in the opposite direction of its rotation.

If you were running in the same direction as the rotation of the ring, it's possible that the sensation of gravity could increase, but this would depend on the specific design of the gravity ring and the rotational velocity at which it is operating. It's also worth noting that running in either direction around the ring could potentially be dangerous, as the high rotational speeds of the ring could cause you to lose your footing and potentially fall off the ring.