Neoliberalism and Postmodernity are inversely related. Both follow what might be called high Modernism, which constituted a set of postwar institutions, values, and assumptions.
Among these were the concepts of the unity of the sciences, technological progress, abstraction in art, the Bretton Woods Accord and the United Nations, mass infrastructure, functionalist architecture, the welfare state, and a globalist outlook. At the same time, however, this unitary view was starkly divided by the the existence of conflicting economic systems in the so-called communist and capitalist realms. Not monotheism, but manicheanism.
In the developed Western nations, the situation produced the "managerial state" stabilizing welfare, Keynesian policies, large-scale public corporations, and broad labor union agreements--all meant to ameliorate the excesses and periodic collapses of capitalism. The Neoliberal revolution grew out of a reaction to inflation, along with a self-conscious conservative-capitalist "revolt." With the decline of state communism, the Neoliberal orthodoxy spread through global deregulation, debt financing, shareholder value, and a positivist, pseudoscientific view of "market pricing" as the sole calculus of value.
While this Thatcher-Reagan-Friedman dogma might claim to be type of social pluralism arising through deregulation and market "freedom," it was in fact a highly idealistic, financially centralized, globally aggressive hegemony imposed upon the world often through violent means. The author you quote seems to be making the point that such "postmodern" economics destroys local values, traditions, cultures, etc. through global finance and rampant commodification.
In a kind of dialectic inversion, this market utopianism gave rise to the various cultural trends loosely called Postmodernism, first in architecture. These are characterized by a skeptical pluralism and critique of the "grand narratives" or "gods-eye-view" of Modernism. The unity of the sciences, for example, could not be naively sustained amid increasing scientific specialization and more attention to the history and actual working methods of scientists. A certain relativism, for better or worse, becomes inevitable with the sheer accumulation of information.
The "inverse relation" I refer to is really the continuing dynamic of capitalism, which at one level unifies mass markets and concentrates wealth, while at another level accelerates the diversification of commodities and the "division of labor," including such cultural labors as science and art. This is undoubtedly an impetus to the various Postmodern trends. The mass-market destruction of local cultures, traditions, architectures, languages, species, and so forth likewise lends urgency to a pluralistic and preservationist reaction.
If you are interested, the geographer David Harvey has written a number of influential critiques of both Neoliberalism and Postmodernism from a Marxist perspective. While Neoliberalism is an active and well-defined ideological program, Postmodernism is more of a set of trends and critiques marked by its skepticism towards such one-size-fits-all programs.