Nate is expected to weaken as it moves inland and could be downgraded to a tropical depression later on Sunday, the Miami-based US National Hurricane Center said.
People in the small fishing municipality of Gutierrez Zamora opened shops and markets in a light rain. Some fishermen had secured their boats on the shore.
The storm was 10 miles (15 km) north of Barra de Nautla in Veracruz, with maximum sustained winds of 45 miles per hour (75 kph), the hurricane center said in its 10 a.m. CDT (1500 GMT) advisory.
Nate is moving west at 9 mph (15 kph) and could dissipate inland by Monday.
Veracruz is one of the top producers of coffee and sugar in Mexico, and flooding could damage the recently-planted crops. High moisture also can cause fungus in coffee beans, and flooding and landslides that cut transportation routes could delay exports.
Sugar crops in Veracruz were greatly damaged last year in an intense tropical storm season.
In addition to the rain, a storm surge will raise water levels by as much as 1 to 3 feet (0.3 to 0.9 meters) above normal tide levels along the coast, the hurricane center said.
The oil-exporting ports of Cayo Arcas remained closed along with two other smaller ports, but Dos Bocas reopened to shipping after four days of inactivity.
State oil company Pemex evacuated 473 workers from five sea platforms as a precaution. Nate had cut Mexican oil production by 178,800 barrels a day as of Friday.
Local media reported a fisherman drowned on Friday after strong surf capsized his little boat in the Bay of Campeche area, and 10 oil industry contractors were missing after they evacuated a rig in the same area.
Meanwhile, Tropical Storm Maria continued on a track away from the Caribbean, where some islands were hit by Hurricane Irene last month.
The storm was located about 50 miles (80 km) north-northeast of St. Thomas in the US Virgin Islands, the National Hurricane Center said in its 11 a.m. EDT advisory (1500 GMT).
Forecast models show Maria, which has maximum sustained winds of near 60 mph (95 kph), moving on a trajectory out over the open Atlantic.
“The center of Maria will gradually move away from the extreme northeastern Caribbean islands,” the Miami-based center said.
