In the neck region between the lobes, the gravity is almost cancelling out. On the other parts of the asteroid, the surface gravity is going to be about $30 \space mm/s^2$. But 216 Kleopatra is spinning really fast, providing up to $20\space mm/s^2$ of radial acceleration due to inertia. $10 \space mm/s^2$ is barely enough to walk. If you are not careful with your steps, you are going to fly for a while before returning to the ground. (Seven minutes airtime for a normal Earth step). You are however not able to throw rocks into orbit, like on Phobos or Deimos, except around the neck where an unstable orbit is just requiring $14 \space m/s$. Just manageable.
As for circumnavigating the asteroid, it is simply going to take too much time. Being patient with the low gravity, moving at say $0.1 \space m/s$, the whole $500km$ trip is going to take about 60 days. Orbiting it is much simpler, only taking a few hours and a minuscule amount of $\Delta v$.
"Down" is going to shift a great deal depending on your location, but your acceleration vector is never going over the local horizon. Otherwise pieces of this rubble pile would have flown into space.