You know, your perception of time passage depend both of internal an external stimulus.
If he, being concealed inside his own dream, only sees white, feels no hungry does not get old, tired or anything inside that ilusion, the brain would simply compact everything (the brain is not a camera, does not register everything, things are registered by relevance, if everything is but white bg, very little of those thousands of years will be registered) and by the 3 or 4rd year or so, he would begin to lose the idea of time passage.
Unless he counts every second, but even then, he would lose the perception of how much time passes from a second to another and it would be meaningless.
So, painfull 5 or so years on a white background, if he does not goes insane due to the sensory deprivation(which requires some unimaginable willpower) before letting go the perception of time, "hes good to go".
Double quotes cuz he would have some mental issues for the rest of his life, but he wont forget anything. Hek, it could even get worse, his hatred that is. One thing is certain, he wont be the same person after this lel.
About the one month, since he is a test subject I guess they will keep him fed. Right? There is no reason to neglect a test subject, since the whole thing about tests, is to know what happens.
TL;DR My point is, if he feels nothing in that white room, how can he feel time passage?