radiation absorption problems

To put your question more correctly, I think you ask what the temperature of the air inside the box will change.

The outer surface of the box absorbs heat effectively because of the black colour. The absorbed heat is transmiited to the inner surface, preferably by conduction. Thus, the temperature of the inner surface rises accordingingly. But because the inner surface is painted silver, heat radiated from it would not be as large as if a dull black surface is used. Heat is transferred more effectively by convection and conduction.

On the contrary, if the inner surface is also painted black, heat transfer by radiation would become more effective. The temperature of air inside the box should appear to be higher.

Be aware that in situation where a very hot object is placed inside the box, the painting of silvery colour on the inner surface helps to preserve heat. It is because heat is now tends to flow out from the box, instead of flowing in as in your example.