I can only comment on Hakone, and I admit I have a bias here as it is where I fell in love with my Fiancé. As you’re already aware, the highlight for many people in this area is to take full advantage of the Hakone Free Pass (about 4,000 JPY for 2 days) and embark on a full tour of the area using the buses, funicular railway, cable cars and (kitsch) boats. I think you can complete the tour in about 4 hours if you want to ‘do’ Hakone. But even from here, there is every chance you won’t see Fuji Sama.
I loved the cable car rides and all the conversations I had on them (as well as waving at never ending streams of passing schoolchildren). I also loved staying and bathing in a traditionally built ex Ryokan (now a more affordable Kokumin Shukusha) and sitting on the balcony in the evening listening to the funicular railway’s musical jingles echoing spectacularly through Gora. And visiting Shinto shrines in the forest glades on the lakesides with the cicadas whooping above us. But these are the kind of pleasures that are best explored with a couple of days. I think if I’d only had half a day to explore Hakone, I would have come away pretty disappointed.