Yes and no, postmodernism certainly embraced the rejection of "first principles" and perhaps elevated it to a new level, but this rejection was neither originated by it nor is specific to it. Traditional justification of knowledge "from first principles" has been a hallmark of classical philosophy, from Plato, to Descartes, to Kant, to Husserl, more broadly it is known as foundationalism. Even empiricist philosophers, like Aristotle, Hume or Mach were in a sense foundationalists, it is just that their foundations included empirical input in addition to (or instead of) the first principles. Foundationalism is roughly a belief in "ultimate grounds" of knowledge, and its breakdown is usually traced back to certain elements of Hegel's philosophy (although Hegel himself was still mostly a foundationalist). It appears more explicitly in life philosophy of Nietzsche and the pragmatism of C.S.Peirce at the end of 19th century, long before postmodernism, and was developed by existentialists like Heidegger and Sartre, and pragmatists like James and Dewey in the early 20-th. The rejection of "grand narratives" is more of a literary gloss on the continued disintegration of foundationalism.
The "data-driven argumentation", if I understand the expression, is argumentation that relies on perpetual empirical (or more generally experiential) self-correction rather than finding secure ground upon which knowledge is to rest, i.e. what is associated with anti-foundationalism. It is aposteriorist and fallibilist about any knowledge, i.e. takes all knowledge, including meta-principles allegedly presupposed to acquire it, to be ultimately derived from contingent (historical) experience and fallible, and hence submits the presumed "first principles" to the same test by evidence as everything else. From this point of view just how grand or localized the narrative should be should not be prejudged. One expression of this trend is the relativization of Kantian "a priori axioms" that pragmatists, and even some phenomenologists and positivists, already advocated, see What are the more complex/interesting examples of synthetic a priori statements? The transition from largely foundationalist to largely anti-foundationalist attitude was boosted by the crises in mathematics and physics in the early 20-th century, up to then the bases and ideals of foundationalist beliefs, but it was slow, both logical positivists and Husserl at the time were still very much committed to foundationalism.
The symbolic watershed is 1950-s when Quine and late Wittgenstein repudiated logical positivism on the analytic side, and Adorno and existentialists repudiated Husserl on the continental side. None of the aforementioned authors were postmodernist, however. The "neo-Marxist repudiation of Enlightenment" is probably an allusion to Adorno-Horkheimer's Dialectic of Enlightenment, which did anticipate many later postmodernist themes, such as cultural relativism, suspicion of "power", critique of "tyranny of truth", etc. But the Frankfurt school, to which both belonged is usually seen as trying to "correct" the ideals of Enlightenment, give them more human "face", rather than to throw them out.
The postmodernism proper is usually associated with French poststructuralism, Foucault, Derrida and Deleuze being early practitioners, but in a more broad sense one can count Kuhn, Feyerabend's "anything goes" and Rorty's "cultural politics in place of epistemology", feminist and cultural critique a la Latour, much of social constructivism, etc., in the same vein. "Grand narrative" is specifically Lyotard's term from much later Postmodern Condition (1979), and is clearly anticipated by Derrida's différance, and Kuhn's paradigm shifts as far as elimination of first principles goes. Or indeed, in reverse chronological order, by Quine's naturalized epistemology, Wittgenstein's rejection of "God's eye view", Peirce's critique of Descartes, Nietzsche's of German idealism, and Hegel's dialectic of perpetual mediation. And while many contemporary philosophers are anti-foundationalists there are alternatives to taking it all the way to post-truth, either culturally or epistemologically, as the continued influence of the Frankfurt school and the naturalized epistemology show.