Sometime after Warcraft III: The Frozen Throne and that fateful cinematic of Arthas donning the now-iconic helm of the Lich King, Chris Metzen made it known: "Arthas and Ner'zhul have become a perfect fusion of one being--Arthas's personality and body with Ner'zhul's wisdom, experience, power, and evil." Indeed, even the events in Christie Golden's novel, Arthas: Rise of the Lich King, indicate that there is no struggle between two entities. Arthas, as we knew him, is dead. Arthas, as we know him now, is The Lich King. And Ner'zhul? He was most likely consumed by the entity which now exists within that armor.
In the quest, Tirion's Gambit, we see how the honorable Tirion Fordring gave up the last vestiges of hope that there is anything left in Arthas to salvage. His words are heavy when he admits, "By allowing me to peer into his heart, you allowed me to confirm what I needed to know. There is no Arthas Menethil anymore--there is only the Lich King."
Even the spirit of Uther the Lightbringer, perhaps one of the greatest paladins to have ever wielded the Holy Light, admits "Arthas is merely a presence within the Lich King's mind--a dwindling presence." When it's considered how long Uther's been trapped within Frostmourne, it's understood that he may have a deeper insight into the situation than any of the more recent heroes.
So it can be safely said that there's a rather strong indication that Arthas, the Lich King, will not find salvation. However, let us discard the above facts for a bit and instead make the claim that there is something left within our wayward Prince of Lordaeron to redeem. Let us assume that there is something of the "good Arthas" left therein. If we make this singular change, what then? Are we still left with an irredeemable villain, or is there a man who deserves the forgiveness than only the Holy Light can give? It's easy to hate the Lich King, after all. He's been one of the most visually iconic villains in Warcraft since Warcraft III. But what about the man who fell in love with Jaina Proudmoore and nearly wed her? What about the young Prince who so dearly loved his kingdom; the strong-hearted son of a king who drove himself mad trying to ensure the safety of his people? Should he be hated with equal passion?
Heavy is the head that wears the crown. Heavy, too, is the head of he who inherits the crown. Arthas Menethil lived in the shadow of his father, King Terenas Menethil II, and wanted nothing more than to do what was best for his kingdom. So passionately did Arthas want to do right by his people that he often allowed his judgement to be clouded. When the Scourge threat befell Lordaeron, Arthas reacted without hesitation--and without consideration of the consequences. He has proven hmself to be a brash man, having made snap judgements that lead to dangerous circumstances. It was this untempered passion that caused him to ignore the obvious dangers placed before him by the enemy. But a weapon untempered is brittle and prone to shattering upon the simplest of blows. Arthas was no different.
Some would claim that Arthas's fall truly began around the Culling of Stratholme, where he lead the slaughter of innocent civilians infected by the plague. Granted, Stratholme was a scenario calculated by Mal'ganis and the Scourge to trap Arthas in a situation he could never win. The young prince had two choices on that fateful day: cull the innocents before they turned to undead, or allow them to become the enemy and kill them regardless. There was no right choice in that matter, as both options lead to the death of innocent people. Stratholme stands as a landmark in Warcraft lore, being the window through which Arthas's allies and friends could catch a glimpse of what was to come.
There are some who believe that Arthas did not truly fall until after he took up the runeblade, Frostmourne, and began to hear the whispers of Ner'zhul. In truth, Arthas's descent began before he and his men ever set foot on the rime covered shores of Northrend. The frozen continent, however, certainly contributed to the prince's fall. It was here that Arthas defied his father's orders to return home, here that he hired mercenaries to burn the ships of his men so they had no choice but to stay, and here that he ordered his men to turn on those mercenaries in vengeance. Indeed, Arthas betrayed the trust of his soldiers and willingly lead them further into the icy heart of madness. His pursuit had ceased being about the people of Lordaeron and had solely become about his own pursuit of retribution.
One can easily argue that Arthas's madness was not his own, but was caused by the Scourge. He became so enthralled by his desire to defeat Mal'ganis that he lost sight of his own humanity. But one must realize that, until this point, there was no magical curse involved in Arthas's descent. The Lich King himself had orchestrated this plan to allow the young prince's own flaws to claw their way to the front. By the time Arthas claimed Frostmourne as his own, he was willing to turn himself over completely to the Lich King's grasp.
Indeed, Arthas Menethil's fall was not voluntary, it was instinctual.
This brings us back to the original question. Now, however, instead of having to forcibly assume that there is still a sliver of the "good" man within the Lich King, we understand with more clarity that he is the Lich King. There is no "good" Arthas for whom we should be seeking salvation, because the Arthas that some people want to see redeemed faded long before Warcraft III ever ended. The blade that was Arthas Menethil's untempered spirit was shattered by the Scourge upon the wastes of Northrend and forged anew in the heart of Icecrown Glacier. As we saw Arthas say in Christie Golden's Arthas: Rise of the Lich King: "Now there is only I. I am the Lich King. And I am ready."
This weapon cannot be unmade, it can only be broken.
Thus, the question breaks down. We can no longer assume that a sliver of the original Arthas Menethil still exists within the Lich King, because the original Arthas Menethil became the Lich King. There is no salvation for one so willingly damned.