When Dumbledore was cursed with Marvolo Gaunt's ring, could he have extended his one-year timer until death with the Sorcerer Stone if it was never destroyed in the first book? Could he have lived longer? Could the power of the Elixir of Life have trumped the power of the curse on Marvolo Gaunt's Ring?
There's a line from Firenze in the first book that implies that Voldemort drinking from the Elixir of Life would bring him back to full strength and undo the curse from drinking unicorn blood.
"The blood of a unicorn will keep you alive, even if you are an inch from death, but at a terrible price. You have slain something pure and defenseless to save yourself, and you will have but a half-life, a cursed life, from the moment the blood touches your lips.”
Harry stared at the back of Firenze’s head, which was dappled silver in the moonlight. “But who’d be that desperate?” he wondered aloud. “If you’re going to be cursed forever, death’s better, isn’t it?”
“It is,” Firenze agreed, “unless all you need is to stay alive long enough to drink something else — something that will bring you back to full strength and power — something that will mean you can never die." (The Forbidden Forest)
There's obviously no canon answer to your specific question, but I think the implication is the Elixir would basically return you to full health even after an extremely nasty curse.
Also note how Firenze phrases it-- that you can NEVER die. Every indication from the books is that the Philosopher's Stone (or more accurately, the Elixir of Life) grants immortality, not just eternal youth.
The ancient study of alchemy is concerned with making the Sorcerer’s Stone, a legendary substance with astonishing powers. The stone will transform any metal into pure gold. It also produces the Elixir of Life, which will make the drinker immortal.
“A stone that makes gold and stops you from ever dying!” said Harry. “No wonder Snape’s after it! Anyone would want it.” (Nicolas Flamel)
"You know, the Stone was really not such a wonderful thing. As much money and life as you could want!" (The Man With Two Faces)
That could be just be clunky writing from JK, but the implication I'm getting is that the Elixir of Life is closer to something like the river in Tuck Everlasting, which prevents death by any means at all.
So the answer appears to be that yes, the Elixir would have saved Dumbledore's life at least temporarily. But I think we also get ample evidence from the books that he never would have done so even if he had the chance.
The Stone had already been destroyed.
"As for the Stone, it has been destroyed.”
“Destroyed?” said Harry blankly.
(Philosopher's Stone, Chapter 17, The Man With Two Faces).
Nicolas Flamel was the only known creator of a Philosopher's Stone.
“Nicolas Flamel,” she whispered dramatically, “is the only known maker of the Philosopher’s Stone!”
(Philosopher's Stone, Chapter 13, Nicolas Flamel).
Since Flamel was dead by the time Dumbledore's hand was affected by the Horcrux curse using a Philosopher's Stone was not an option. The only Stone had already been destroyed.
According to the timeline of the story, therefore, Dumbledore using a Philosopher's Stone to fight the Horcrux curse isn't possible. What would've happened in a what-if scenario is impossible to say.