If we go with what is canon in the Buffyverse, Spike with a soul is an entirely different person than the soulless monster who kill for over a century, including two Slayers, tried to rape a third, and bragged about being the "Big Bad." By that interpretation, ensouled Spike has nothing whatsoever to atone for.
If we ignore that canon, then Spike should certainly regret "his" past, but also learn from it, vow never to become that monster again, and devote the rest of his existence to making the world at least a little better by fighting evil and aligning himself with the forces of good. (pretty much what Faith did when she decided not to return to prison after Season 7) Nothing short of time travel could ever erase the crimes that the monster committed, or bring back the lives that he took. The past is past. Brooding, weeping, and gnashing of teeth don't accomplish anything useful.
Spike can't change the past. It's over and done with. He CAN, however, try to become what Buffy saw in him: a better man, and by so doing, help to make the future a little better. That's just about as close to actual atonement as he can get.