Does the white hole behave as a source of “anti-gravity”, in other words?
No, you’d still have to exert thrust away from a white hole to keep still, because accelerations are unchanged by time-reversal. And if you stop thrusting and let yourself fall towards a white hole, you would always continue moving towards its horizon, but you would never pass through, you would just get asymptotically closer and closer. This is the time reverse of what happens to things that escape from close to a black hole (under inertial motion, by virtue of having a large initial outwards radial velocity): they were never inside the horizon, but if they were arbitrarily close to the horizon and moving away from it with sufficient speed they will eventually escape — but it will take a very long time if they were very close to the horizon.
