What is postmodernism in context of art criticism? I know that postmodernism rejects the idea of universal truth and in context of literary art criticism it shifts the focus from the writer or author to the audience - but then, what tools of art criticism does it provide the reader and the art critic? I mean, does it entail relativism so that 'anything goes'? Does postmodernism dismiss the entire idea of art criticism?!
I would not say that postmodernism is well-defined in any field and tends to resists such definition. I believe the first applications in criticism were in architecture and simply referred to the technical and aesthetic options available beyond "modernism," a specific period which was fairly well defined by, in a nutshell, "form following function," or at least an auspicious absence of ornamentation and representation.
So the "postmodern" in architecture might include Venturi's return to representation or drawing on historical aesthetics. Yes, in a sense "anything goes." Painting isn't as capital intensive as architecture, so the transition from the "high modernism" of Rothko or Klein through Rauschenberg, Lichtenstein, or Warhol to "postmodern" has a lot more hybrid types. But once again, it dispenses with the "rules" against representation or mixed styles, say, along with the modernist idea of some sort of necessary progress or, as they say, "grand narrative."
A lack of overarching standards is rarely a problem for critics, and while I can't tell you what exactly might constitute a school of "postmodern" art criticism, I am sure there is plenty, drawing on a big grab bag of theories, from semiotics and Marxism to feminism. My guess is that this is all very liberating to criticism, since writing about political content, social criticism, painterly style, theories of representation, or the more philosophical approaches of Danto is all on the table.
So the style of criticism, like the styles of painting or architecture, is open to pitched battle in the so-called marketplace of ideas, including political and social critique. Uninteresting, irrelevant...those might be the most damning criticisms in this milieu. Again, the idea is that no larger sense of historical progress dictates the values and "critique" will therefore generally be internal to the aims and purposes of the work, not to some transcendent standard.
I would identify a pair of tendencies, seemingly opposed but often welded together: on the one hand I would submit the idea that we cannot confront the real without some kind of subjective dissolution, I.e., the truth is a matter of how much an individual can ‘take’ or accept the world, and that taken ‘unvarnished’ it would even be harmful, so that art is necessary to protect us from the truth (we could read this as a necessary prophylactic function of social fictions, enabling us to forget certain difficult things in order to enjoy ourselves; or on the other hand, aesthetic production as an endless series of masks in order not to die of boredom, banality and brutality of actually-existing society).
On the other hand, the idea that truth in itself doesn’t exist, but is a matter of interpretation and creative selection, amid endless shifting perspectives reflecting culturally-specific embeddings. The reality that exists ‘in between’ perspectives is here the subject of aesthetics; there is not a single shared life world unifying all of society, but rather a universal emptiness that cuts across every existential order in the form of irrational antagonisms and hegemonic struggles for social control. The Cartesian subject who would still ‘want’ the truth in the postmodern situation is required to make an act, an ‘ungrounded’ decision as an embedded agent with insufficient norms and knowledge to guide them, in order to ‘fill in’ the gap of their subjectivity with the positive order of a cause that transcends them: to make a creative leap is always to dare to decide an undecidable situation, with necessarily insufficient grounds...
It is very common to propose postmodernism as the new hegemonic metanarrative, come to replace the old one - but the approaches grouped under the term are opposed to that entire concept of methodology, at a core level. It is precisely not the point.
Many prototypical 'postmodernists' like Foucault, never accepted the term applied to themselves, never defined themselves by it, never felt it to be a cohesive school or movement. And, really, it isn't.
Postmodernism arose significantly through literary criticism. It is better understood as a toolbox, for trying to see the cognitive biases of candidate texts, and critiquing the non-explicit aims embodied by them. Foucault is frequently criticised as having no concept of truth separate from power - he absolutely did, but his focus & interest was on how implicit power plays are embodied in our ideas.
A great next step is understanding Kuhn. As a theorist of science as a whole, eg compared to Popper, he is really problematic. He has no methodology for truth, there is only culture, and 'combat' within the culture's rules, the new paradigm 'winning' the new hegemony and providing status and power to those who brought the paradigm shift about. But as part of a toolbox, he is saying, don't pretend scientists are 'free floating brains', they are part of culture too, science is culture, and though it is not fundamentally subjective (as, sadly, Kuhn did actually claim, in his hubris), we still do have to consider human needs for belonging and solidarity to understand how science progresses. See the replication crisis, & many other science scandals.
Baudrillard has a narrow focus on abstraction, how culture shifts to simulation and simulacra. Again, the tendency is to approach his methodology as the new metanarrative, but it's an area of attention, and set of tools for understanding the preconceptions and biases we have, inherit, and embody as culture.
I would relate postmodernism to art criticism as the shift from what art IS (implicitly a metanarrative, making a claim to be the dominant interpretation in cultural discourse), to what art DOES, for it's different audiences, for it's maker, considered in different contexts, in layers of abstraction and reference. Art as text.
This doesn't mean no argument is better than another, or there is no such thing as a better informed commentator. But, that the arbitration in a given context is by pursuasion. And crucially, always subject to revision. A perspective may be considered very compelling, definitive even, in one era, but be revealed later as relying on implicit assumptions - like European bias against Africa. Later generations just found that obvious, that pre-WW2 Europe could not see.