While Nietzsche undoubtedly paints himself as an enemy of the left, he is certainly no friend of capitalism.
For Nietzsche, capitalism represents a system of bourgeois values. Insofar as capitalists see the value of human beings, not in their free spirit, great deeds, or artistic creations, but simply in terms of their ability to manage investments, capitalist values are directly opposed to Nietzsche’s thought.
Nietzsche sees the ideal class as an aristocracy: a group of people freed from the toil of labor by the sweat of those less excellent, a “high society” where greatness and free spiritedness can prevail (albeit it for the few).
Nietzsche believes that capitalism creates a completely decedent society (the worst form of society so far, in fact). Against the adolescent reading of Nietzsche, his thinking vehemently opposes so-called “libertarian” philosophies (such as those espoused by Ayn Rand, Paul Ryan, and other teenagers).
Consider this passage from The Gay Science (aphorism 40):
Soldiers and leaders still have far better relationships with each other than workers and employers. So far at least, culture that rests on a military basis still towers above all so-called industrial culture: the latter in present shape is altogether the most vulgar form of existence that has existed.
Remember this is the late 19th century, the most laissez-faire era of capitalism. He continues:
Here one is at the mercy of brute need; one wants to live and has to sell oneself, but one despises those who exploit this need and buy the worker.
This even echoes Marx in Capital (in criticism, if not in answer). Yet Nietzsche continues:
Oddly, submission to powerful, frightening, and even terrible persons, like tyrants and generals, is not experienced as nearly so painful as is this submission to unknown and uninteresting persons, which is what all the luminaries of industry are. What the workers see in the employers is only a cunning, bloodsucking dog of a man who speculates on all misery; and the employer’s name, shape, manner, and reputation are a manner of complete indifference to them.
Suffice it to say, Nietzsche hated capitalism, thinking it the worst form of existence so far. Under tyranny, one at least has some great general or leader as a boss. Under capitalism, the greediest buffoon (a person completely uncultured and without taste except for profit) takes command.
While Nietzsche’s ideal form of government certainly isn’t socialism (and indeed these reactionary elements of his thought are troubling), it’s quite easy to demonstrate that the “libertarian” appropriation diametrically opposes his thought, so much so that Nietzsche would prefer a completely militarized society over a laissez-faire democracy.