I don’t know about the cost, but other answers here are saying it can’t be done - it has actually been done already. ICBMs are designed to move a lot of mass in ballistic trajectories- but the same technology is easily adapted to send rather lower masses to LEO and beyond. They can be converted to send small payloads to places as far away as the Moon at least. The Dneper is a lightly converted Russian ICBM which has been used for many satellite launches under an agreement to let it use some of its decomissioned ICBMs for peaceful use. It has the capability to send 550 kg on a translunar injection orbit which means it could crash that much onto the Moon and not that much more delta v for orbit around the Moon. I don’t know what that translates into for a lunar soft landing.

Lift-off of the Dnepr (rocket) - a Russian ICBM lightly converted for peaceful use.

As for humans - it can send 3.2 tons to the ISS. That’s three times the mass of the orbital module for the Soyuz TMA and almost enough to include the re-entry module as well. Soyuz (spacecraft)

So in principle, it can launch the mass of a human occupied spacecraft to LEO - if you were really desperate - but it would be dangerous - not human rated, humans need much lower failure rates unless you are willing to take a significant risk of death - and also - need for reliable life support etc. But in terms of mass - yes it could take a human + life support to the ISS, and if you sent the re-entry vehicle separately they could return as well. So if you had no concerns at all about safety, you could keep the ISS occupied with humans entirely using ICBMs I think. But don’t recommend it!

USE OF ICBMS FOR SMALL ASTEROID DESTRUCTION

Note that one Russian researcher has proposed an idea to use modified ICBMs as a last minute defense against smaller asteroids. The big advantage for this is that they use solid fuel and are designed so that they can be launched at a moment’s notice, unlike conventional rockets. If you had only a day or two, or a few hours warning, you couldn’t even launch a conventional rocket in time, but could launch an ICBM easily. However to blow up even a 50 meter diameter rock, I think they have nuclear detonations in mind.

It could be a reason to develop or retain nuclear weapons in a peaceful civilization that would never think to use them against its own citizens. Maybe many ETIs do have nuclear weapons for defense of their planets. But not so sure it is a good idea for Earth. I don’t think it is strictly speaking against the Outer Space Treaty myself, as what makes it a weapon of mass destruction is if it is intended for killing humans in large numbers. If the only intent is to destroy an asteroid to save humans - is that a “weapon of mass destruction”? It seems rather a “weapon for prevention of mass destruction” so long as it is used as intended. And it wouldn’t be stationed in space, only launched at need.

Anyway this is the idea here: Russia Wants to Turn Old Missiles Into an Asteroid Defense System

The smallest nuclear weapon the US has acknowledged existed weighed only 23 kg and had up to 1 kiloton yield. So given that the Dneper can take 550 kg to the Moon, does seem you could probably get ten 1 kiiloton nuclear weaons to an incoming asteroid on eacha ballistic missile. Special Atomic Demolition Munition

They didn’t actually say that they planned to use nuclear weapons, and ICBMS converted to use conventional explosives would be much less controversial. More speculations about the idea here Russia to modify Cold War missiles to destroy asteroids

If the relative velocity is kilometers per second perhaps just kinetic impact of 550 kg is enough? I don't know what you'd need to break up a 50 meter diameter asteroid. Many of them are probably "rubble piles" loosely held together. Perhaps a million tons for 50 meters diameter. 550 kg impactor so about half a ton, at say 20 km / second, that's enough to divert a million tons at about 1 cm / second. In theory that's plenty to separate out the pieces by of order of a kilometer or so by the time it gets to Earth, separations of kilometers. So it might not take much depending on how strong it is and how many pieces you can break it into, but there again it might not work at all. I think they'd need to test it first and there are plans to do that.

If we had good space telescopes such as the Sentinel, half a billion dollars, then we wouldn’t need this before long, as it would map nearly all of even the smallest asteroids within a decade or so, and if you know about the asteroids a decade or more in advance they are easy to deflect, no need for nuclear weapons. See my Giant Asteroid Headed Your Way? - How We Can Detect And Deflect Them