Colonizing the asteroids starts at home!
When I first heard of this proposal I was reminded of the recommendations of the Space Studies Institute to do ore processing experiments on orbit as the first step to building O'Neill colonies. They were talking about using Lunar regolith simulant in low Earth orbit, with the goal of developing the techniques to utilize the products of a future lunar mining operation, but as an asteroid resources advocate I'd always preferred to think about doing the same thing with a captured asteroid.
A few years ago I wrote about colonizing a near-Earth asteroid (without moving it), with a focus on artificial gravity issues. The reality is that we don't yet know enough about the composition of any asteroids to have a decent shot at making water, oxygen, plant nutrients, or any of the other things you'd need for a space colony. We need to learn it before the colonists are sent, and having a captured asteroid to experiment on is a great way to do it.
Ultimately, though, the largest asteroids you can capture won't be big enough for a colony. Designing a mission to take a few hundreds of people out to (at least) 20 lunar distances is quite a challenge if you want to get them there healthy and ready to build a new world. Most problems can be solved by throwing mass at it, but typically that means more launches. Having material that is already in orbit, especially high lunar orbit or the Lagrange points, which you can use for shielding, know how to process into consumables or even structural components, would be of great help.
How likely is survival of that first colony? Mostly, they'd be cut off from resupply - close approaches to Earth of the same asteroid only happen infrequently. If they are to survive and thrive they'll have to be independent and stubborn. They'll need to see what they're doing as important and have a forward looking motivation.
After just a generation, they may be ready to expand. If their home is an Apollo asteroid, they might have the option of hopping over to the asteroid belt. Their well-honed technology will come in handy there. Eventually they may pull apart entire asteroids to make O'Neill type colonies, or just very large spaceships.
Visiting Earth will be easier for the asteroid dwellers than visiting the asteroids is for the Earthicans.