I thought Alex was faking his cure, but I thought his sickness was real when he heard the 9th. I believe Alex was faking because before he signed up for the treatment he was the model prisoner, and wanted out, even if it meant he would have to stop his violence. I also think the fake burp is very note worthy, as well as the minister choosing Alex because he was enterprising.
The reason why I think the music made him sick was because it was so near and dear to him. He literally got off on the 9th, and would celebrate his violent evenings by listening to the 9th after he got home. After the treatment, he wanted to fake being cured, but he couldn't fake the fact that he loved the 9th and associated it with ultra violence, which he could not commit anymore. Also, if the treatment really did work, why would it just be that song that caused a negative reaction and not other songs?
I think given the choice; Be yourself and stay in jail, or be fake and be free was too much for Alex...which caused him to jump.
As far as him "being cured" at the end...I just simply can't understand it. They hint at the fact that he had a lobotomy...but I don't think he was finally "cured" until he spoke with the Minister, and it was conveyed to him that they want him to go back to what he was...for their own political survival.
I really don't know to be honest. But this is the take I had on the film. And I've never read the book, so there is that too.