The story is a well-known piece of Jewish folklore, and it is hard to know if the story is true at all, and if it is, how much it may have changed over the generations.
The story can be found in the sefer Michtavei Chofetz Chaim (p.262) at the beginning of an essay written in honor the shloshim of the Chofetz Chaim. It was originally published in the periodical Baderech. The author uses the pseudonym מחר הלוי.
The following is the version of the story printed there:
על הגר"א מוולנא יסופר, כי הוא פנה פעם למוכיח-בשער ידוע בימיו: הטף לי דברי מוסר! המוכיח הזדעזע: האם הטף אטיף דברי תוכחה להגאון מוולנא?! אני גוזר עליך שתעשה את זה - צוה הגר"א. ופתח המוכיח ואמר דברים אלה: הוי, אליהו, אליהו, עובד אתה את בורא-העולם בהיקף מוגבל, בסתר חדרך הצר ובד' אמות שלך, - צא נא לאויר העולם, לשוק החיים והמעשה, ועמוד שם בנסיונות יום יום, ועבוד שם את האלקים מתוך כשלונות היצר הרע ומלכודותיו על כל צעד ושעל... התחיל הגר"א בוכה ואומר: להיות עושה נפלאות ומהלך על חבל נטוי על פני ים זועף - אני ירא
Informal translation:
It is told of the Vilna Gaon (Rabbi Elijah of Vilna), that he once turned to a mochiach (literally, "admonisher", referring to a rabbi who would give public speeches admonishing the people) who was well-known in his day and said, "Tell me some words of mussar (moral admonishment)!" The mochiach was shocked, "Shall I give mussar to the Gaon of Vilna?!" "I decree upon you that you do this!", commanded the Vilna Gaon. The mochiach began and spoke these words, "Woe, Elijah, Elijah! You serve the Creator in a very limited manner, hidden in your room and within your own four cubits. Go out into the world, into the public arena of life and action, and withstand the challenges of daily life, and there serve God amidst the setbacks and traps of the evil inclination at every step... The Vilna Gaon began to cry and said, "To do wonders and walk on a rope stretched out over an angry sea - I am afraid!"
As you can see, while not exactly the same in every detail (esp. in the fact that the baal mussar is not identified), this is clearly the same basic story. Personally, I suspect the story is either entirely fictional, or has changed substantially from its original form.