Imagine having two boxes side by side out in the sun. Both boxes have cardboard as the floor and side surfaces. The top surface of the box is a piece of glass to allow in heat and insulate it. Each box contains a thermometer. If one box has the inside of it lined with mirrors (inside walls and bottom surface) and the other has the inside of the box lined with black paint (inside walls and bottom surface), which box will be hotter on the inside if left out for 5 minutes, as an example?
When a photon interacts with an atom, three things can happen:
elastic scattering, the photon will keep its energy and change angle
inelastic scattering, the photon will give part of its energy to the atom and change angle
absorption, the photon will transfer all its enregy to the absorbing atom (electron)
You are saying out on the sun, but let's just assume first we are talking about visible light, that interacts with the surfaces. Higher energy photons will go deeper into the material of the box and will do 2. inelasstic scattering, but it is not clear from your setup how thick the material of the box will be.
The black inside of the box will do 3 mostly, that is absorption. Most of the visible light will be absorbed by the black surface's atoms. In this case most of the visible light's energy will be transferred into the vibrational energy of the atoms and molecules of the black surface material (heat up).
Now the mirrored inside wall box will do mostly 2., that is inelastic scattering, where the energy of the photons will be kept, so mostly no energy will be transferred to the vibrational energies of the surface of the material of the mirrored inside walls. Now that is not completely true, because photons do exerpt pressure on the walls of the inside of the box, and transfer some of their momentum to the walls. Still, most of the photons' energy will be kept and will eventually (since the walls inside are mirrored) leave the box back to the outside via the glass on the top (I assume the glass is both ways seethrough).
So eventually the black inside box will have higher temperature.