Understanding the intricate interplay of mechanical forces and fluid dynamics at the cellular scale is crucial for unravelling the complexities of brain function. Notably, cellular swelling and volume regulation modulate brain states [1], while mechanical forces play a pivotal role in structural plasticity and learning [2]. Recent advances in high-resolution electron microscopy reconstructions of the mammalian brain pave the way for detailed numerical investigations into these physiological processes in detailed cellular geometries. Addressing the computational challenges posed by the high geometrical complexity of entangled brain cells, we propose efficient and scalable solvers for cell-by-cell models of cellular mechanics and fluid flow in cerebral tissue. Employing a dual-poroelasticity approach, we represent both the intracellular and extracellular spaces as poroelastic media. The elastic solid models the cytoskeleton and the extracellular matrix, respectively, while the fluid network represents intra- and extracellular fluid. The permeable cell membrane separates both domains and allows hydrostatic and osmotic pressure-driven fluid exchange, resulting in a Robin-type interface condition. We present discretization techniques and parameter-robust, norm-equivalent preconditioning strategies for the resulting interface coupled poroelasticity problem. Specifically, we focus on the geometric setting of a large number of cells embedded in the extracellular space, which gives rise to a solution operator with a nontrivial near nullspace. Demonstrating the efficiency and scalability of our proposed solver, we present numerical results showcasing cellular swelling on large-scale 3D reconstructions of the rat visual cortex.