The spacecraft would first cast an extendable tether toward the object and attach itself using a harpoon attached to the tether. Next, it would reel out the tether while applying a brake that harvests energy while the spacecraft accelerates. This allows Comet Hitchhiker to accelerate and slowly match the speed of its ride, and keeping that slight tension on the line harvests energy that is stored on-board for later use, reeling itself down to the surface of the comet or asteroid. A comet hitchhiker spacecraft can obtain up to ~10 km/s of delta-V by using a carbon nanotube (CNT) tether, reaching the current orbital distance of Pluto (32.6 AU) in just 5.6 years.
Unfortunately rocket scientists apparently don't read the opinions of Anonymous Cowards on the internet, or else they'd know from discussions last year that it simply won't work. It seems that the idea defies "basic orbital mechanics" and "makes no sense".