Assuming charitably that the missile could generate an intercept, you would probably get some sparks, maybe more or less depending on how much of the rock is actually nickel and iron. But basically, the rock is a rock. An AIR-2 Genie likely wouldn't bother it all that much beyond maybe a surface glazing and some neutron activation. The most a Sidewinder will do is expensively nothing.
If they can bust few (double digit?) meters of concrete, they could bust a rock. Assuming they dont miss and assuming they dont slip.
https://en.m.wikipedia.org/wiki/Bunker_buster
Obviously it would have to be some combination of an anti sat missile (so it can fly in space) and a bunker buster (so it can crush throigh rock).
I'm not sure that counts as shooting it down. In the context of a celestial object usually "shoot it down" means "deflect it so it misses earth".
F15s could fire anti satelite missiles in early 1980s. Who knows what "toys" are available 40 years later.
The benefit of this sort of task is in both cases the target object doesn't move of its own accord, so presumably if you can predict the path of the object with enough precision then it would work. Clearly we can pinpoint a 1m satellite, but no idea on the error bars for a 1m meteoroid.
A small nuke could definitely obliterate a 1-meter rock, if you don't mind contaminating a large part of Europe with radioactive dust.
[0] https://en.m.wikipedia.org/wiki/Category:Nuclear_anti-aircra...
Not sure if they arent much bigger than an anti sat missile though.