In this paper, we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions. Moreover, if we further assume that the volume decays at most polynomially at a singular point, then we obtain a H\"older estimate of the holomorphic function at that point. In a similar spirit, we also established a continuity estimate for bounded harmonic functions, with a finite dimensional exception, at a singular point of a manifold satisfying a weak mean value property and a polynolmial volume decay condition.