I think at some point, Arthas would kill Muradin and claim Frostmourne anyways.
However, this is after all a "What If" so, once Arthas reached the shores of Northrend, he would probably find that Muradin was already fighting the Undead with all the strength of his newly acquired blade. Afterwards he along Arthas had defeated the Undead, Muradin would probably vanish much like Arthas did after he killed Mal'Ganis. Corrupted by the blade, Muradin would return to Ironforge, an honorable welcome awaited him, but he moved simply into the throneroom, thrusting Magni in his chest, thus the threat of the Dwarves would be over for the Lich King.
After this, Muradin would probably follow the story Arthas followed in RoC, as he would first be visited by Tichondrius and do his bidding. Though, since the nation of Lordaeron still has their king and Lightbringer, they would perhaps once more be able to counter the Scourge of Lordaeron. It is hard to know, but if Lordaeron fell once more, perhaps Muradin would be the one who would "lure" Illidan into claiming the Skull of Gul'Dan, perhaps it is he who will become One with the Lich King.
