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Many viruses must enter the cell nucleus through small nanopores in order to replicate. We model here the viral motion as a stochastic process described by the Survival Fokker-Planck equation. We estimate the probability and the conditional mean first passage time that a viral trajectory is absorbed at a small nuclear pore before being terminated. The method is based on the explicit Neumann-Green's function. The cell nucleus is modeled as a three dimensional ball, covered with thousands of small absorbing windows. The minimum distance between them defines the smallest spatial scale that is an unavoidable limit for efficient stochastic simulations. Derived asymptotic formula agree with stochastic simulations and reveal how small and large geometrical parameters define the cytoplasmic stage of viral infection. [ABSTRACT FROM AUTHOR]