The speed of sound in graphene is 22 km/s. If your tether were made of monocrystalline carbon fiber, it looks like you would have to boost the whole apparatus to 8+ km/s first to catch a 30km/s object, and you could then leave the solar system at something less than 52 km/s.

For comparison, the New Horizon probe to Pluto and beyond left at 23 km/s.

With a thousand km of graphene-fiber tether standing perpendicular to the object's path, it would take 45s to accelerate the probe at a peak of 484 km/s/s. For a 10kg probe, that would put 4.8e9 N of tension on the tether.

Graphene has a tensile strength of 130000 MPa, or 1.3e11 N/m^2, so the cable would need to be some 20cm thick.

Boosting to a higher speed would reduce the cable thickness needed proportionally. At 19 km/s, quite doable, it's 10 cm. However, boosting 1000 km of 10cm cable to 19 km/s would take quite a fair bit more fuel than just boosting the probe itself to the target speed of 41 km/s.