For a coastal city like New York the obvious choice is a Tsunami. These can be triggered a number of ways (Earthquake, Volcanic explosion, Landslip, Meteor strike in ocean) and even a high tech first world country would only get a few hours warning depending on where the trigger event happens.
Read up on Krakatoa for the volcano cause, Japan recently had an earthquake-caused one. You can look at La Palma in the Canary Islands for the Landslide trigger although there is some debate as to just how large the Tsunami would be.
This would involve a new hot-spot appearing in the crust and a volcano forming off the coast of New York (which would take several years at least). The side of that volcano would then collapse letting sea-water in. The resulting explosion and tsunami could then be devastating and that final act would happen very quickly.
The pressure wave generated by the colossal fourth and final explosion radiated out from Krakatoa at 1,086 km/h (675 mph).It was so powerful that it ruptured the eardrums of sailors on ships in the Sunda Strait, and caused a spike of more than 2½ inches of mercury (ca 85 hPa) in pressure gauges attached to gasometers in the Batavia gasworks, sending them off the scale. The pressure wave radiated across the globe and was recorded on barographs all over the world, which continued to register it up to 5 days after the explosion. Barographic recordings show that the shock wave from the final explosion reverberated around the globe 7 times in total. Ash was propelled to an estimated height of 80 km.
Even in non-geologically active areas earthquakes can still occur, although they are rarer. In order to generate a large tsunami a moderately sized earthquake would normally be combined with an underwater landslip as happened in Japan.
They later found evidence on the seafloor of a landslide with a horizontal footprint about the size of Paris, measuring 40 kilometers by 20 kilometers (25 miles by 12.4 miles), and 2 kilometers (1.2 miles) thick. The landslide created a focused tsunami wave that, when combined with the power of the earthquake, reached epic proportions, he said.
Some islands, particularly volcanic ones, have vast amounts of rock and earth in potentially unstable positions. A collapse in that island that happened fast enough and in the right direction could potentially trigger enormous tsunamis.
However, the western half of the volcano has an approximate volume of 500 cubic kilometres (120 cu mi) and an estimated mass of 1.5 trillion metric tons (1.7×1012 short tons). If it were to catastrophically slide into the ocean, it could generate a wave with an initial height of about 1,000 metres (3,300 ft) at the island, and a likely height of around 50 metres (164 ft) at the Caribbean and the Eastern North American seaboard when it runs ashore eight or more hours later.
A meteor impact into water would in many ways be more destructive than one into land. Huge waves would radiate out in all directions causing the coast to be battered by at least one and possibly multiple (as the waves bounce around) tsunamis.
A computer simulation of an asteroid impact tsunami developed by scientists at the University of California, Santa Cruz, shows waves as high as 400 feet sweeping onto the Atlantic Coast of the United States.
Man Made: (See Sheraff's answer for more details)
If too much were extracted from the wrong parts of continental shelves then that could in itself trigger the underwater landslide that then triggers the Tsunami. For example Methane Hydrate extraction in the wrong places could cause just this scenario to happen.
If methane hydrates were ever extracted from continental shelves without appropriate precautionary measures being taken, the disaster scenario painted by Schätzing in ‘The Swarm’ could occur: without methane hydrates, the shelves on the coasts could become unstable, gigantic landmasses could start to slip and that in turn could trigger tsunamis. Experts are taking these risks seriously and are only experimenting on the extraction of methane hydrates in flat storage locations far from continental shelves.