We don't know.
Since this version of reality only appeared in the single episode Turn Left, we have basically no knowledge about it other than what's stated explicitly in that episode. All we know about Captain Jack Harkness in the "turn right" version of the universe is the following:
ROSE: It's the ATMOS devices. We're lucky, it's not so bad here. Britain hasn't got that much petrol. But all over Europe, China, South Africa, they're getting choked by gas.
DONNA: Can't anyone stop it?
ROSE: Yeah, they're trying right now, this little band of fighters, on board the Sontaran ship. Any second now.
(The sky burns.)
DONNA: And that was?
ROSE: That was the Torchwood team. Gwen Cooper, Ianto Jones, they gave their lives. And Captain Jack Harkness has transported to the Sontaran home world. There's no one left.
-- Series 4 Episode 11, Turn Left (emphasis mine)
Note that it doesn't say he was kidnapped by the Sontarans, nor that he "was transported" to Sontar (which would suggest against his will), but that he "has transported".
Perhaps he went to Sontar with a honking great gun, using the teleporters that we saw on the Sontarans' spaceship in The Sontaran Stratagem and The Poison Sky, and committed himself to destroying them or at least ensuring they wouldn't be able to come and attack Earth again. A massive great punch-up between Jack Harkness and millions of Sontarans would be a sight to see, and I'm sure both sides would have a great deal of fun.
Perhaps the Sontarans did kidnap him so that they could try to turn him into a weapon they could use in their fifty-thousand-year-long war against the Rutans. An immortal warrior would be a great asset for them if they succeeded in recruiting him.