Ms Cherry was accompanied by her parliamentary aide at the time.
It is understood no weapon was involved and there was no violence.
The man is understood to have called at Ms Cherry’s constituency office the day before and been told of surgery times.
Police made an arrest and Ms Cherry continued with her surgery.
A spokesman for Ms Cherry said: “We were very impressed with the professional way in which the police responded.”
A Police Scotland spokeswoman said: “Police in Edinburgh have arrested and charged a man with a breach of the peace after being called to Thorburn Road around 11.45am on Friday.
“He is expected to appear at Edinburgh Sheriff Court on Monday.”