It seems to me that there are two ways of thinking about this, one resulting in an answer of "yes" and the other resulting in an answer of "no". Both ways require defining what "round" means.
One definition, I think, is to mean encircling an object. If that is the case, then the man encircled the pole which the monkey was on, therefore the man has encircled the monkey, and the man has gone "round" the monkey. So the answer is "yes".
However, if you restrict that same definition to also require exposure to all sides of the object (Front, back, left side, right side), then the answer would be "no" because the man is never exposed to any side of the monkey except the front.
I agree with the first definition I proposed more than the second definition because the first definition requires less "defining" which seems to follow the simplicity of the question asked. (Similar to some kind of Occam's Razor mentality... Maybe?)
A satellite in an orbit around the Earth that matches the Earth's rotation (so that the satellite is always over the same part of the Earth) is known as a geosynchronous orbit. Orbit being defined as, "a path described by one body in its revolution about another". A satellite in geosynchronous orbit, orbits, or goes round the Earth. Likewise, the man is going around the monkey.
Not sure yet I am willing to hear others' opinion. For me this question can't be discussed at all. I will be here to thank anyone who has given the answer and a lot of cast-iron reasons in order to back up their argument. Anyone has the right to answer.