That includes the $114 million in federal reimbursement money that's being released now, Cuomo told a Sunday afternoon news conference at the East Side complex.
The first overnight patients are expected at NYU's main Tisch Hospital section on Dec. 27. This week, Grossman said they expect only "ambulatory" patients admitted for brief stays.
The rest of the $1.1 billion will come from insurance and "more FEMA money," Grossman said.
Cuomo said he believes hospital staff who quickly evacuated the patients had "saved people's lives."
The governor said he was in Manhattan on the night when Superstorm Sandy waters flooded the neighborhood, "and I saw the East River blending into Second Avenue."
During the surge, 324 patients had to be evacuated, Grossman said.
Grossman called the new federal funding "a shot in the arm" of his medical center, which he said would reopen to all services by the end of January. Some in-patients will start being admitted this coming week, Grossman said.
Richard Cohen, vice president for facilities operation, said air pumps were inserted into low-lying areas of the hospital that had been flooded to fully dry them out.
Cohen said power and heat has been restored to several of the buildings at the medical center, which "is almost open for business."
The governor said long-term reconstruction for the city after Sandy will takes "months and months."
He and other New York officials have requested about $32 billion in federal funds for the state. Cuomo said Sunday that the White House has presented the bill to Congress, which must approve before more money is allocated.
"We are urging them to act quickly, before the end of the year," said the governor. "We need it passed, we need it desperately."
The governor said he and other New York officials "are working as hard as we can" to procure more federal funds for all four hospitals that were hard-hit by Sandy and forced to close.
Three other hospitals were closed to inpatients in the aftermath of the storm. About 700 patients had to be evacuated from Bellevue when all but one of the hospital's generators failed. The other two medical facilities are the Manhattan VA Medical Center and Coney island Hospital.
Coney Island Hospital has partially reopened and is due to be fully operational by January. Employees from the Manhattan VA Hospital, which remains closed, have been relocated to the Brooklyn VA Medical Center.
The money for NYU Langone happened to be released first for various bureaucratic reasons beyond the state's control.
Cuomo said he's pushing hard "to cut through the red tape."
Earlier Sunday, the governor visited Long Beach off Long Island's south shore, where the storm wiped out roads, beaches and homes.
Long Beach is getting more than $24 million. Almost $16 million goes to Nassau County Public Works.