I'm not sure this totaly meets your requirement for a passive system, but I think its at least part way there and is a novel approach so worth a mention
What you need is a “vanguard” Think a very wide ultra-thin circular membrane together with minimal stabilisation and communications equipment traveling a long way ahead of the generation ship on exactly the same heading. If the vanguard is sufficiently far ahead there would be time for it to detect any rock passing through its membrane and give sufficient warning to the generation ship to fire emergency lateral thrusters to adjust course slightly and avoid the object.
A variation on this would be a small fleet of small probes fitted with wide angle laser projectors and detectors plus minimal communication and stabilization equipment. These could also fly far ahead of the generation ship in a wide circle and create a net of laser light that could detect any rock passing through and warn the generation ship in a similar way. If any of the probes were hit themselves the others could take over giving multiple redundancy. The other advantage would be the capacity to have a much wider net giving a greater safety margin.
Extended answer
Assuming a hole in the pressure vessel which is survived but leaves the ship damaged and the crew trying to figure out what happened.
First I would assume they would find the hole quickly as there would be emergency procedures in place to detect and deal with such an eventuality. After the immediate aftermath the hole would be visually examined and that would give them a lot of information.
They would know if it had been an explosion from within the ship outwards, or a hit on the ship inward and in cases of external impacts the location of the hit on the ship and the exact time would give away the direction of approach. The size of the hole would also tell them something about the energy of the impact and possibly even the mass and velocity of the projectile. There is an outside chance that they might even be able to find fragments. Considering it approached from behind it can impact at relatively low or relatively high velocities.
If you need further sensor confirmation there might have been cameras on the hull which would have detected the impact flash. The other possibility (if approaching from behind) might be interference with communications from Earth at a very low level or general scientific sensors measuring anything you want on an ongoing basis in all directions continuously. If it was metallic it might have made some record in magnetometer readings during its approach.