True.

Chester A. Arthur was indeed the third man to serve as President of the United States in 1881. When that year began, Rutherford Birchard Hayes held the highest political office in the US. After the inauguration of the new President in March, James A. Garfield was at the helm until he was shot on July 2nd. President Garfield seemed to improve from the wounds he sustained as the summer progressed, but he succumbed on September 19th. The very next day, Vice President Chester Alan Arthur was sworn in as the 21st President of the United States.

Thus, 1881 began with Hayes as US President. After the Inauguration in March he was succeeded by Garfield, who served as the Chief Executive until passing away on September 19th. The day after Garfield died, Arthur assumed the presidency.

More Info: en.wikipedia.org