He does not seem to have been interested in his father's business; instead he spent much time travelling. He kept a diary, and that became the material for a book published after his early death: Henry B. Smith, A Memorial of Anson G. Phelps Jr., New York 1860.
On a steamship passage from Naples via Malta and Greece to Constantinople in March, 1841, he met another traveller, Hans Christian Andersen, who in fact recorded Phelps's name in his diary. However, Andersen had not yet achieved world fame (none of his writings had at the time been published in English), so it is doubtful whether Phelps realized that he was in the company of a great writer.
The business of Phelps Sr. was continued by his two sons-in-law, see http://www.raken.com/american_wealth/early_industrialists/greene_phelps3.asp