After Neo's body is flushed out of the pod and the Nebuchadnezzar lifts him out with the crane, you can see that he's taken out face up. However, when he's about to go in the ship, the crane is holding him face down. Although you can't see his face, you can tell because you can see his knees (not the back of them).
No, he is still facing up, but it's just the camera angle at first that makes it look like he's facing down. If you look closely at the very end of the scene as he's being pulled into the light, you can see that his knees are hanging down, which yould make him be facing up.