For spotting Mt. Fuji you need a bit of luck. The iconic mountain is known for being surrounded by clouds more often than not. Locals even call Fuji “shy” for that.
The visibility problem is to some extent man-made, actually. There are a couple of big paper mills alongside Fuji River, discharging large amounts of vapor. Chances to get a glimpse of Fuji are best after some strong rainfall, when the haze is washed away from the air.
Mt. Fuji is just 100km distant from central Tokyo. With its view across the skyline of Shinjuku, the Bunkyo Civic Center is one of the city’s more exciting vantage points. The building’s top-floor observation platform is open Mon–Fri, 9am–8:30pm; admission is free.