I don't know of any old inns in old Japanese buildings (Minka) in Hakone. (I'm sure someone here will jump in if there is one). There are some very high-end ryokan, though. (And the prices reflect that). Personally, I'd forgo Hakone in favor of a really rustic spot, but if you want to wake up to a view of Mt. Fuji, then Hakone is one place to do it.

If I were you, I'd probably take the train rather than flying to Kyoto. And I'd probably try to fly out of KIX to Germany rather than backtracking to NRT.