Krdytkyu

An asteroid approaches and the Moon 'catches' it in the same way that a sports player catches a ball - that is to say by matching the velocity of the hand to that of the ball.

Could a lucky slingshot approach cause this to happen?

Assuming that the Moon has no atmosphere, my intuition tells me that there must be a direction and velocity such that an asteroid can do this. Does mathematics say otherwise?

Gravity assist

In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft.

NOTE

I mean that the asteroid approaches the Moon in a direction opposite to the Moon's Earth orbit. It then starts to swing around behind the Moon and just happens to contact the Moon's surface at the time the exact matching of speed occurs. Thus landing relative to the surface at zero velocity in any direction.

In order to land on the Moon, you must, at some point, be moving towards the Moon (decreasing your distance from it, to be more precise, you may also be moving sideways) and close enough that the Moon's gravity dominates that of the Earth and the Sun. From that point on, your kinetic energy (relative to the Moon's centre of mass) can only increase as you get closer to it (you are converting potential energy to kinetic), so gravity cannot slow you down. So you will always crash at roughly the Moon's escape velocity (2.3 km/s) or more. You might crash straight in, or graze the surface while moving almost horizontally, but there will be a substantial relative velocity.

@SteveLinton's answer is right, no matter how gently you try, by the time you get to the surface the Moon's gravity will have accelerated you to something like 2,400 m/s. There are ways to use the gravity of the Earth and Sun to make a tiny reduction in this, but it's a very small effect.

The simplest way to argue this is that rocks on the Moon don't suddenly, spontaneously jump up and fly into deep space. Classical mechanics works basically the same forwards and backwards in time (in lossless systems as @Mołot importalty points out). So if something can not happen in one direction in time, it can't happen in the other either.

In theory a soft landing is possible, although not with our moon.

As others have said, the rock inherently comes in at a minimum of escape velocity. You can't actually spin the moon that fast or it would fly apart--but what if your object is a solid piece of rock spinning at a high rate? It barely grazes the moon, it's spin makes up for the lack of enough spin of the moon. It bumps lightly and eventually rolls to a stop. (Note that it will have to be spinning far above it's escape velocity and thus must be held together by chemical bonds, not gravity.)

I don't know the details, but in the right configuration, if two bodies that are orbiting each other approach a third body, one of the bodies can be captured into orbit around the third body and the other body is ejected from the system. A considerable part of the kinetic energy acquired by one body from gravity is eventually carried away by the other.

Perhaps an extreme case can be constructed where the captured body is actually left at zero orbital velocity on the surface of the capturing body.

As another answer notes, classical mechanics is reversible, so the reverse sequence is that an incoming body passes near the Moon, gravitationally picking up and carrying away a mass sitting on the surface of the Moon.

I'm going to take a different approach here. Let's suppose that such a rock does in fact land on the surface of the moon. The reason I do this is to show a more substantial reason why it cannot happen.

We know to begin with that the only forces that could be acting upon the meteor are gravity and electromagnetic forces. We know this for a fact because it is in the vacuum of space and those are the only two forces that could feasibly interact without a medium. We know that it must be moving towards the surface of the moon or have a component of motion in that direction. The reason for this is that the meteor moves towards the moon and will slow down as it reaches it. Since we are only considering the moon and the meteor we know that the moon has to be pushing on the object so as to decelerate it. We will assume that motion tangential to the ground on the moon is somehow matched by trajectory. This is possible since the object could have a tangential motion speed equal to that of the moons rotation and that speed wouldn't change. However it would get faster as it approached the surface in the sense that it has to travel a greater distance to keep up when on the surface vs in orbit. So we mean rotation speed of the moon at the surface and not in orbit. This also means that the exact speed is determined by the precise landing location's elevation. Since there are many craters on the moon and it is not perfectly flat this does make the situation less likely.

However, now you have to stop and think about this for a moment. The moon is pushing the object away in such a way that it decelerates it. Now unless this force causes the meteor to fracture at the surface, this means that the moon can push the object away and there is nothing that will stop it from immediately doing so. Meaning that the meteor will hit the moons surface very lightly (can't be exactly at 0 velocity or it would stop midair) and then start being repelled by the moon and be launched into space.

However, this situation will not occur because the moon is not magnetically charged as far as I am aware, and furthermore such a planet with a charge greater in magnitude then the force of gravity would likely be unstable and start repelling itself. The force of gravity is what we primarily see as the strongest non-contact force in the universe specifically because (barring theoretical dark matter) it does not cancel.

The only other alternative is that something external to the moon pulls on the meteor causing it to decelerate and that said object slowly stops pulling on the meteor such that the acceleration is almost 0 at the surface of the moon. However your scenario does not include such parameters so I cannot justify it as being a valid case.