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Cellular scale decision-making is modulated by the dynamics of signalling molecules and their diffusive trajectories from a source to small absorbing sites on the cellular surface. Diffusive capture problems which model this process are computationally challenging due to their complex geometry and mixed boundary conditions together with intrinsically long transients that occur before a particle is captured. This paper reports on a particle-based kinetic Monte Carlo (KMC) method that provides rapid accurate simulation of arrival statistics for (i) a half-space bounded by a surface with a finite collection of absorbing traps and (ii) the domain exterior to a convex cell, again with absorbing traps. We validate our method by replicating classical results and verifying some newly developed boundary homogenization theories and matched asymptotic expansions on capture rates. In the case of non-spherical domains, we describe a new shielding effect in which geometry can play a role in sharpening cellular estimates on the directionality of diffusive sources.