Cardiac arrhythmia causes significant mortality, and to understand its mechanisms and design effective treatments, numerical modeling is necessary. Homogenized models, such as the bidomain model, represent the action potential at tissue level, but do not describe arrhythmogenic cellular-level details such as fibrosis or ion channel distribution. We consider a recent cell-by-cell model with explicit representation of myocytes. Due to their size, such models present new simulation challenges concerning aspects of mesh generation, time stepping, and preconditioning. We will present a geometry and mesh generation approach to define large-scale computational meshes as well as efficient adaptive time stepping and domain decomposition preconditioning. Tetrahedral meshes are based on implicit surface-meshing of a random network of cells, designed to mimic real cardiac tissue, and so far extended to represent 400 cells in 15M tetrahedra. Despite the pronounced locality of solution features, the overhead induced by spatial adaptivity has proven to outweigh the efficiency gains. We present a combination of spectral deferred correction (SDC) methods as higher order time stepping and computationally inexpensive algebraic adaptivity, which can be interpreted as multi-rate integration and speeds up cardiac simulations both in monodomain and cell-by-cell models significantly~\cite{ChSW2022}. The approach uses nested subset selection on the algebraic level and explicit embedded error estimates, and incurs only negligible overhead. The dominant diffusion and the elliptic constraints of the cell-by-cell models require implicit time-stepping methods and an appropriate preconditioning of the arising linear equation systems. We will present a novel domain decomposition preconditioner of BDDC type that is tailored towards the special structure of the cell-by-cell models, and discuss its convergence properties both on analytic and numeric ground. We demonstrate overall performance and convergence rate through a combination of SDC, algebraic adaptivity, and BDDC preconditioning on 2D and 3D examples.