Notes from the Underground is generally seen as a forerunner to existentialism, so the text presents many of the philosophy's main tenets, if in a less developed way. Let's start with absurdity: the Underground Man's logic is twisted and convoluted, often contradictory. He concludes that the universe is without reason. There's also a strong current of radical, existentialist freedom and responsibility: the Underground Man suggests that man has the freedom to fight against everything, even 2+2=4, but he also admits that he is to blame for everything in his life. Because he is always alone, the Underground Man acts as a great forerunner to the existentialist notion that every man is in constant isolation from his fellow human beings. Last is the claim that there are no primary causes of the things we do, hinting at the later existentialist belief that man always creates his own meaning through action.
The Underground Man fails to recognize the Existential solution to his "inertia" dilemma: there may not be good reasons or justifications for the things we do, but we all must act anyway.