When T. S. Eliot says "to spit out all the butt-ends of my days and ways.", he is alluding to cigarettes. The butt-end of a cigarette is the part that isn't smoked is spit out. He his comparing his life to a cigarette butt because he feels as if his life has no meaning or is worthless and no one really cares about it.

T. S. Eliot is alluding to William Shakespeare's tragedy Hamlet. In Hamlet, Prince Hamlet is a ferocious and somewhat reckless hero. He likes to take matters into his own hands, evident by his pursuit to avenge his father's murder. By saying he is not Prince Hamlet, he is saying that he much prefers to sit back and follow directions. He is most certainly not elegant or reckless like Hamlet.