Q:

Did Voldemort believe that the Elixir of Life could save him from the curse of drinking Unicorn blood?

In Philosopher's Stone, Voldemort says:

"See what I have become?" the face said. "Mere shadow and vapour...I have form only when I can share another’s body...but there have always been those willing to let me into their hearts and minds...Unicorn blood has strengthened me, these past weeks...you saw faithful Quirrell drinking it for me in the forest...and once I have the Elixir of Life, I will be able to create a body of my own...Now...why don’t you give me that Stone in your pocket?"
(Philosopher's Stone, Chapter 17, The Man With Two Faces).

Did he think that the Elixir of Life could remove the cursed life that drinking unicorn blood brings?

A:

His action proved that he believed in it. There are details in the book which prove it.

“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.”
Harry Potter and the Sorcerer's Stone, Chapter 13, Nicholas Flamel

Firenze believes Elixir of Life will resolve the issue with the curse of drinking Unicorn blood

“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. Mr. Potter, do you know what is hidden in the school at this very moment?”
“The Sorcerer’s Stone! Of course — the Elixir of Life! But I don’t understand who —”
Harry Potter and the Sorcerer's Stone, Chapter 15, The Forbidden Forest

And finally Voldemort himself accepting it.

“you saw faithful Quirrell drinking it for me in the forest…and once I have the Elixir of Life, I will be able to create a body of my own….Now…why don’t you give me that Stone in your pocket?”
Harry Potter and the Sorcerer's Stone, Chapter 17, The Man With Two Faces

