Does anyone know, when an electron in an atom is exited by absorbing a photon, how long does it take for the electron to absorb the entire energy of the photon? Does the time scale depend on the energy level?
It is not that simple, mainly because the only thing we can calculate is the probability to find the electron in a given state. If you drive a two level system (e.g. an atom where the energy of the photons are chosen so that only one transition is active) using e.g. a laser what you find is that this probability changes a rate proportional to the amplitude of the drive field and the dipole moment of the transition(meaning yes, it does depend on the levels); this rate is just the inverse of the Rabi frequency.
Note that the main difference between the "semi-classical" Rabi model and the QED version (using a J-C Hamiltonian) is just the presence of the vacuum Rabi oscillations in the latter; i.e. the semi-classical model with a classical drive field is actually quite good.
I know that this is not what you asked for, but the point is that Rabi rate is the only time-scale we can actually talk about or indeed observe in experiments.