Police say they received a report about the stolen truck around 5:30 a.m. and determined that it was the vehicle involved in a collision at 7th Avenue and 14th Street.
Two parked cars were also struck.
The bus driver, 49-year-old William Pena of Hillside, New Jersey, was pronounced dead at the scene.
The driver of the truck, Dominic Whilby of Griffin, Georgia, was taken into custody and treated for non-life threatening injuries at Bellevue Hospital.
Three others, including two bus passengers and a coffee cart vendor, suffered minor injuries.
Whilby reportedly had a 0.18 blood alcohol reading, twice the legal limit. The assistant district attorney told the judge that he "was in New York for less than half a day before he began his reign of terror on the citizens of New York County."
Whilby is charged with manslaughter, vehicular manslaughter, grand larceny, criminal possession of stolen property, criminal trespass, criminal mischief and felony assault.
Increased charges are also being considered. The judge ordered him held without bail.
Whilby will return to court on February 18.