Does the 1 train take you to 123 William Street in Manhattan?

No. You have to take the 2 or 3 trains to Fulton Street (at William Street). The address 123 William Street is on the same block as the subway station, between Fulton and John Streets.

You can transfer from the 1 to the 2 or 3at Chambers Street, 14th Street, Penn Station, Times Square, 72nd Street, or 96th Street.