I say we catch that sucker, tow it into our neck of the woods as a second moon, and use it as a space port. Are there a million problems with this proposal? Yes. But would it be totally rad to do this? Also yes.
> At about one-quarter the diameter of Earth (comparable to the width of Australia), it [the Moon] is the largest natural satellite in the Solar System relative to the size of its planet, the fifth largest satellite in the Solar System overall, and is larger than any dwarf planet.
It would be a hell of a world to go from "desert" to "so we've got all these new oceans to name and also you can't breathe it but ground level pressure is one atmosphere".
But sure if it’s gonna happen, let’s get the neighbors a pool! Just let me uh… change my shorts.
An entire fragmented comet impacted Jupiter in 1994
> Over the next six days, 21 distinct impacts were observed, with the largest coming on July 18 at 07:33 UTC when fragment G struck Jupiter. This impact created a giant dark spot over 12,000 km (7,500 mi) across, and was estimated to have released an energy equivalent to 6,000,000 megatons of TNT (600 times the world's nuclear arsenal).[24] Two impacts 12 hours apart on July 19 created impact marks of similar size to that caused by fragment G, and impacts continued until July 22, when fragment W struck the planet.[25]
> Although the impacts took place on the side of Jupiter hidden from Earth, Galileo, then at a distance of 1.6 AU (240 million km; 150 million mi) from the planet, was able to see the impacts as they occurred.
We’re getting better at this stuff!
(It's not possible in any reasonable time)
Highly impractical, and would take centuries to slow it down enough.
For a setting where we were getting it right until we got it wrong, see Niven's "A World out of Time".
What this means is that if do you have a way to move it into a circular orbit around the sun, you could gain energy from the process, rather than having it cost energy. Maybe the best way to gather that energy, and possibly to move it as well, would be through gravitational assists, since they are lossless kinematic interactions. The energy gathered could then be used to build more of whatever is doing the interaction, exponentially speeding up the process.
Calculating the exact path of an orbiting body under thrust is difficult. But the object will be within 1% of the desired orbital radius for roughly a 22-degree arc of its orbit centered on the periapsis, during which time its path will be closely approximated by a circular arc 648 million km long.
Let's say that over the course of this arc, we want to slow it from its initial speed of 12.7 km/s to the required circular orbital speed of 9.0 km/s. That means we need a continuous deceleration of roughly 0.00006 m/s^2 over a period of 2 years.
Assume that we'll produce this thrust by launching material from the object into space using mass drivers. By the Tsiolkovsky rocket equation, the smaller the fraction of the object that we want to use as reaction mass, the larger our "exhaust" velocity has to be. If we want to only lose 10% of the total starting mass, we need our exhaust velocity to be about 10x the total desired delta-V -- that is, 37 km/s, or roughly 0.01% the speed of light. This is a tall order, but let's say we can somehow solve the engineering problems and build a linear accelerator that can get rocks moving that fast.
Assume the object is 160 km in diameter and made entirely of ice, giving it a total mass of about 2.0e18 kg. The total required momentum change is therefore about 7.5e21 kg m/s, and the required energy input is 1.4e26 J. If we assume constant thrust for two years, this means we would have to launch about 3.2 million tons of material per second, averaging out to 2.2 exawatts of power required.
To put this number in perspective, it's several million times higher than the average electricity generation of the entire planet Earth. To generate this much power using 100%-efficient solar panels, at a distance of 11 AU from the sun, you would need a solar array approximately half the diameter of the sun itself.
So a direct propulsion approach, at least, doesn't really seem like it's within the realm of feasibility.