A simple example of quantum transport in a classically chaotic system is studied. It consists in a single state lying on a regular island (a stable primary resonance island) that may tunnel into a chaotic sea and further escape to infinity via chaotic diffusion. The specific system is realistic: It is the hydrogen atom exposed to either linearly or circularly polarized microwaves. We show that the combination of tunneling followed by chaotic diffusion leads to peculiar statistical fluctuation properties of the energy and the ionization rate, especially to enhanced fluctuations compared to the purely chaotic case. An appropriate random matrix model, whose predictions are analytically derived, describes accurately these statistical properties.