One can be a pragmatist (albeit, perhaps, not an instrumentalist) even without giving up the idea of approaching the truth. Popper was not the first philosopher who tried to reconcile radical fallibilism with a form of realism. Many ideas of scientific methodology associated with him, including the "hypothetico-deductive method" and falsification, were developed decades earlier by the founder of pragmatism, Peirce, see SEP on Peirce's logic of science. Quine also advocated a form of fallible realism, albeit less robust than Peirce's, or even Popper's. Quine is sometimes called a "logical pragmatist", but this is using "pragmatism" broadly. Like Popper, he was more influenced by positivists than by classical pragmatists (Peirce, James and Dewey), but he was more of a pragmatist than Popper, who remained much closer to positivist optimism about the reach of formalization and the theory/observation distinction. It is unclear how much Popper, positivists or even Quine were familiar with Peirce's works, their publication stretched over decades after his death.
A common theme to such reconciliations is that reality is posited as hypothetical with its features determined through some form of inference to the best explanation. Roughly, positing the ontology of our best scientific theories as real/existing best explains their empirical success (Peirce adds that Humean empiricist view of laws of nature defeats science as an explanatory enterprise, an ethical point touching on intellectual values). What is also common to Peirce and Popper (but not Quine) are the partitioning of reality into different kinds of existence, and the idea of convergence of scientific knowledge to "the truth". Peirce had an ontology of acting laws and relations, both inside and outside the "subject", underlying the derivative objects, both concrete and abstract, existent and real-possible, and Popper had his three worlds. According to Germino:
"“World 1” represents physical objects and states; it is the “total world of the materialists.” “World 2” consists of states of consciousness and subjective knowledge. “World 3” represents the “whole world of culture,” or knowledge in the “objective sense”... Popper calls the third world the realm of “epistemology without a knowing subject. Not surprisingly, Popper attempts to positivitize, as it were, the contents of his third world, calling it the realm of “objective knowledge.”"
The thorniest issue, that both Peirce and Popper struggled with, was to reconcile the historical and cultural dependence of fallible knowledge with postulated mind independence of the "objective" reality that it is supposed to "approximate". And here their paths diverge.
Their ontologies are very different, and so is the idea of "approximation". Popper mostly followed in the footsteps of logical positivists by looking for measures of corroboration and approximation such as degree of verisimilitude, "truthlikeness". The more true sentences a theory entails the more truthlike it is, see SEP's Truthlikeness. Even aside from unpleasant formal consequences (no falsehood can be as close to the truth as a logical truth, a triviality, and no false theory is closer to the truth than any other) Popper's account uncritically imports the notion of truth itself. It seems at times that he wanted to keep the traditional correspondence truth, common to "folk philosophy" of scientists, along with the falsificationist epistemology. Peirce was much more sensitive to Kantian problems that such a mix poses for a realist. Particularly, to unifying conceptual accounts of diverse cognizing subjects into a single reality, and describing subject's interaction with it, see What are the similarities/differences between how Kant thinks 'noumenon' limits understanding compared to C.S. Peirce?
This caused his much greater departures from the correspondence truth since for Peirce "objective knowledge", being an abstraction, is not of a kind with reality itself, a living acting entity interacted with by a living acting subject. This is the core of Peirce's pragmatism, which is absent in Popper and is somewhat subdued in Quine. Peirce builds an additional epistemological superstructure of habits of action and concepts as conditional forks in them, a pragmatist semantics, and an elaborate theory of signs, his semiotic, to connect abstractions to reality through action and reaction and account for what is now called the problem of intentionality. This is absent in positivists, but then they were not realists, but it is also absent in Popper, whose attempt to cash in "convergence to truth" via formal measures was not very successful. In fairness, many find Peirce's justification of mind independence and convergence to the truth to also be a stretch, but at least he paints a more plausible picture of cognitive processes and epistemic growth.