Yotsuya station to Shinjuku station is 5 minutes on the JR Chuo line (orange train) so I hardly think it's worth it to change where you are staying to get closer to Shinjuku.
Furthermore, that Oakwood Shinjuku is really closer to Okubo station. To walk from that Oakwood to Shinjuku station would take as long as taking the Chuo line from Yotsuya (the sign out front of that apartment says Shinjuku station is about 630m whereas Okubo is around 350m away).
If I were picking the nicest Oakwood location to be at, I'd pick Azabu Juban in a flash.