To the west in Puerto Rico, government offices closed early, adding to traffic jams as islanders headed home or sought emergency provisions.
One death was reported in the neighboring island of Culebra. Authorities say a 55-year-old man collapsed from cardiac arrest while trying to install storm windows on his house.
In the British Virgin Islands, residents flocked to supermarkets for supplies. To protect against deadly shocks from downed power lines, the electricity will be turned off across the territory once winds reach 40 mph (65 kph), said government spokeswoman Sandra Ward.
"Hospitals are in emergency mode," Ward told The Associated Press. "They have generators and confirmed they have backup generators filled with diesel and ready to go."
In St. Croix, the Hovensa LLC oil refinery, the second-largest in the Western Hemisphere, was shutting down until after the storm passes, said spokesman Alex Moorehead. St. Croix is the most-populous of the U.S. Virgin Islands with more than 50,000 people. It stood in the direct path of the late-season storm, which could grow into a Category 2 storm before driving over the islands after midnight.
Hurricane warnings were also in place for Antigua, St. Martin, St. Kitts and Nevis and other islands.
The storm forced diversion of Carnival's Triumph cruise ship to Jamaica and Grand Cayman, away from stops in St. Thomas and San Juan, spokeswoman Joyce Oliva said. Carnival's Glory ship canceled a stop in St. Maarten on Wednesday and will instead visit Grand Turk.
Authorities in Puerto Rico prepared shelters in Vieques and Culebra and planned to evacuate elderly residents from the two small islands off the east coast of the main island.
Omar was a Category 1 hurricane with winds near 85 mph (140 kph).
The National Hurricane Center in Miami said Omar was expected to plow over the northeastern Caribbean islands then head into the central North Atlantic, well away from the U.S. mainland.
Omar's center was located about 235 miles (380 kilometers) south-southwest of St. Croix and 235 (380 kilometers) miles south-southwest of St. Croix, at 2 p.m. (1800GMT) Wednesday and was moving northeast near 13 mph (21 kph).
Meanwhile, another tropical depression was hugging the coast of Honduras, and a tropical storm warning was in effect for the area.