There was me thinking he just took advantage of something not part of his plan.:/ That's how I read that particular part of the game, so it's not bullsh!t to me.
Of course Arkham's endgame was to screw Vergil over, that was obvious from how he talked to Vergil. But I didn't think he figured getting stabbed into the deal. The most he mentions during his big reveal as Jester was that he used that disguise to get them into that room to use their blood.
The stabbing arises because Vergil is disgusted with Arkham for being weak enough not to kill Lady. I don't think Arkham would have predicted Vergil would react that way and consider him no longer useful, not when they were nearly at the lair of judgement. That was the endgame.
He didn't need to be stabbed to get Lady in that room. If he really needed to, Arkham could've just gotten Lady to chase him in there dressed as Jester.
Or maybe she could have found the room by herself?:/
Besides, the dialogue before Arkham was stabbed was Arkham being all excited about reaching the lair of judgement. He thought he'd won, that he'd gotten them where he wanted them. Heck, Vergil cuts Arkham off in the middle of speaking when he stabs him.
Arkham thought he was useful enough to Vergil so no harm would come to him. Also, at some points in the game, it's Arkham telling Vergil what to do. Like when Vergil wants to finish Dante at the top of the tower, but Arkham tells him to leave it, and he does.
So, considering that, I think Arkham thought he was safe from being stabbed by Vergil.
But in the end, it's two guys using each other as tools, pretending to be allied until they can get what they want. They deserve each other.