Could a rugged spaceship haul an asteroid "moon" back to Earth?

Asteroid...definitely the smaller ones. Moon...not likely unless talking about the hundreds of mini-moons that are cropping up almost daily as being discovered around some planet or another.

We have the technology for and have actually remotely closed in on comets, which are way harder to track than asteroids. Deep Space 1, an ion driven rocket, closed in to asteroid Braille and Comet Borrelly and almost, but not quite landed on the comet..all by remote control and AI software. [See source.] DS 1 was a tiny rocket, but give us something a bit bigger and we have the technology to scoop up a small chunk and bring it back to Earth.

So the size of the spacecraft's hold would be the major limit. Towing it back would be a bit difficult as controlling the craft with that much mass hanging onto its tail would be problematic. But hauling internally can be and has been done with the shuttles; so that's a non-issue.

If the spaceship were big enough and powerfull enough, and the asteroid were small enough, then yes.

A moon is not an asteroid.

asteroid

noun

1 a small rocky body orbiting the sun. Large numbers of these, ranging in size from nearly 600 miles (1,000 km) across (Ceres) to dust particles, are found (as the asteroid belt) esp. between the orbits of Mars and Jupiter, though some have more eccentric orbits, and a few pass close to the earth or enter the atmosphere as meteors.

moon

noun

• a natural satellite of any planet.

O.K., I reread your question and decided to give a bit less detail. It's not impossible, but it would take such tremendous scientific development that it isn't likely that the human race will live to see it happen. You would have to have an EXTREMELY large ship to not be pulled in by the gravitational attraction, and you would have to exert a force that is most likely stronger than the attraction between the Sun and Earth. Even if you could, you would have to deal with the possibility of the thing orbiting the Sun rather than the Earth due to the fact that the Sun would have a higher attraction than that of the Earth depending on the distance.

It would be a fairly simple matter, with an active space program, to direct things like Apophis into one of our Lagrange points. It would be fatal to move in something really large, because it can get unparked fairly easily.

An no, the object does not need to be inside the space ship. You could affix engines and fuel system to the rock and blast away. You could use a "gravity tractor" if it were a loosely packed ball...