The following section of the Marvel's Spider-Man guide answers the question whether there is a fast travel system in the game. The mechanics of Spider-Man movement has been perfectly reproduced, but sometimes you have to get somewhere fast - taking a short break from swinging on the web.
Answering the question quickly - yes, in the game you will experience fast travel. The game's action setting, New York, has been divided into nine districts. In each region there are police stations and special buildings like F.E.A.S.T or Peter Parker's apartment. Simply click on the location icon on the map and confirm the fast-travel option. A short cutscene awaits you, in which your hero will move to his destination by a subway.
The fast journey is available quite quickly, right after you start one of the main missions - "Home Sweet Home". Additionally, when you visit a research station for the first time, you will be able to get to it from the map screen.