The IPM near Earth is 5 particles/cm^3. If we assumes this is 5 hyrdogen atoms we won't be far wrong.
First, lets do some Feynman math to see how plausible this is.
Everything is about as dense as water, give or take an order of magnitude. A water-density object that weights 1000 tonnes (or $10^6$ kg) is going to be about 10 meters on a side.
Solar escape velocity at Earth Orbit is about 40 km/s. So your asteroid will pass over a volume of 10 m * 10 m * 40 km / s or 4 * 10^6 m^3/s.
There are 100^3 cm^3 in a m^3, so this is 10^6 * 4 * 10^6 * 5 hydrogen atoms/second, or 4*10^12 hydrogen atoms/second.
There are 6*10^26 hydrogen atoms per kg, so this is 0.7 * $10^{-14}$ kg/s of hydrogen swept up.
Times two, times c^2, gives us about 1.25 kwatts.
The kinetic energy of a 1 kilotonne object moving at 40 km/s is 8*10^20 J. The gravitational binding energy of such an object is 8 J. This is a problem.
Assuming the energy released is 100% efficient at slowing the asteroid, it would take about a million light years at that speed to reduce the kinetic energy of this asteroid by 50%. Collisions with the ISM won't, on average, do anything detectible other than make the asteroid glow with high-energy photons (and deposit some heat).
If the asteroid hit something more substantial (say, a mote of dust or whatever), the first problem is that anything that can slow it down from solar escape velocity will also blow it apart.
The problem is that the binding energy of any natural asteroid is going to be tiny, and the KE of a comet is huge; any impact that is enough to soak the KE in a short period of time is going to shatter the asteroid.
Can chemically bound might survive such an impact? A pure anti-Tugsten has a chemical binding energy of 850 kJ/mol. That mol has a weight of a bit under 200 g. 1/2 (40km/s)^2 * 200 g is 160,000,000 J. So an impact that would slag the asteroid would shave off half-a-percent of the KE of the asteroid.
So no impact/ism based energy deposit is going to do anything useful.
For the asteroid to be captured, it will have to get extremely lucky with orbital mechanics. It falls inward, does a close pass by a gas giant that "slingshots" it. That might be enough to kick it into a long elipitical orbit. If the slighshot (through insane luck) then swung by another large planet "just right", it could shed more KE and have a less elongated orbit.
A traditional capture would involve doing this dozens of times over millions of years, where a a few failures results in the asteroid being ejected from the system or hitting a planet.
In your situation, the best you could hope for is losing 5-10 km/s from a gas giant flyby. It has to get lucky and avoid hitting anything larger than a mote of dust (even a mote of dust will create an explosion, but maybe not enough to destroy the asteroid).
Then you might be able to use a gravity tug, over decades, to move it into a better orbit.
Once you have it reachable...
To cut is, fire a stream of electricaly neutral particles (to avoid giving it a net charge; any electron you shoot at it will cause a random proton to be attracted to the object as it seeks to become electrically neutral). As even a ghostly amount of such particles hit, it will produce an extremely vigorous energy emission; far more than anything you could generate for anywhere near the same amount of effort. I might fire separate proton and electron beams to cut different spots, or have them merge into a neutral beam.
Once you have defeated whatever chemical bonds are keeping the asteroid together, they'll drift apart easily.
You won't want to get anywhere near the object. Even rocket exhaust should be avoided near the target; it will be insanely more dense than the ISM. So you might want solar sails, avoid being either up or down "wind" of it (downwind, because it will shed antimatter; upwind, because you'll shed matter). You'd cut a piece off, then can move that piece by hitting it very gently with a beam.
Capturing an isolated piece becomes extremely tricky. Tricks with magnetic bottles etc are fine, but probably the easiest is to throw near-vacuum matter at it and let it glow and move itself.
You'd maintain some kind of defence system around the anti-asteroid, to avoid even dust particles from hitting it. As noted, a few microgram dust particle is enough to give it a kick enough to utterly change orbits; a defence system powerful enough to detect [b]every little bit[/b] of microgram dust and deflect it would be an extreme challenge.