It wasn't really a challenge. He stormed into solitude and used a shout to kill the high king instantly. Then he fled on a horse. The solitude gate was opened by Rogvir, i believe was his name. He was the one being executed the first time you visited solitude.
I know, but if you talk to the court wizard in Solitude, she says when Ulfric came we thought he only wanted an audience with the High King.  If we had only known ... but it was to late.  Ulfric challenged the High King to a dual.  To refuse would give Ulfric reason enough to call a new moot to decair a new high king, so he had to accept.
It was a fair fight, all things considered.  Ulfric just new the rules of the game and had a wild card hidden away. 
And that's why everyone in Skyrim is so confused right now.  Nords are undergoing a transformation.  They were the barbairian race when things started back in arena.  Rough, strong, mead drinkers who would go on adventure and smash some skulls rather than stay at home or deal in the world of politices.  But slowly, they've become more and more involved with the politics of Tamirel.  They're becoming "modern" in that worlds sense.  Thats why many nords fear there way of life is going away.  Tradition vs Modernisation.  The Nords as a people don't know what is more important to them.  That's why this happened the way it did.  Imperial Law had been slowly working its way into Skyrim, but at the same time it's traditions of the strongest leading were still valid.  The new and old layers of the government were rubbing against one another, and it finally caught flame with the banning of Talos, Skyrim's "Son".
And that's why I really like Skyrim.  My fav race is changing, like a real race of people would.  Both sides are in the right, both morally and in terms with their law.  And in he end, the side that is "evil" will be the side who lost and gave up afterwards.  
Because it's not the victors who write history, but the survivors.