On April 4th, a new bus terminal named the "Busta Shinjuku" will be opening adjacent to the Southern Exit of the Shinjuku Station. All long distance buses operators such as Keio, JR and Willer Express will depart from this new terminal effective April 4th, including the buses headed to the Mount Fuji area. The Airport Limousine Bus for Narita/Haneda Airport will also depart from the new terminal. The buses will leave from the 4th floor while there will be a taxi pool on the 3rd floor.
At the moment, buses leave from various points around Shinjuku, so the move will make it far easier for travelers to connect from trains to buses. So far, there is not much info in English other than the bulletin on the Willer Express site (http://willerexpress.com/st/3/en/pc/information/89.php) It might be a good idea to arrive at the terminal a few minutes early as the details on where each operator will set up their check-in counters haven't been announced as of writing.