which methods were used for construction?
Different methods were used such as cut-and-cover being one of them
New York City Subway
When the IRT subway debuted in 1904, the typical tunnel construction method was cut-and-cover. The street was torn up to dig the tunnel below before being rebuilt from above. Traffic on the street above would be interrupted due to the digging up of the street. Temporary steel and wooden bridges carried surface traffic above the construction.

when was the subway built in new york In 1904
wiki:19918501
New York City Subway [SEP] Construction methods
Street railways had to be torn up to allow the work. The foundations of tall buildings often ran near the subway construction, and in some cases needed underpinning to ensure stability. This method worked well for digging soft dirt and gravel near the street surface. Tunnelling shields were required for deeper sections, such as the Harlem and East River tunnels, which used cast-iron tubes. Rock or concrete-lined tunnels were used on segments from 33rd to 42nd streets under Park Avenue; 116th to 120th Streets under Broadway; 145th to Dyckman Streets (Fort George) under Broadway and St. Nicholas Avenue; and 96th Street and Broadway to Central Park North and Lenox Avenue.