The fastest scenic option
The Odakyu Romancecar Limited Express runs direct from Shinjuku Station to Hakone-Yumoto in 85-90 minutes (¥2,470 reserved). Most properties offer a free shuttle from Hakone-Yumoto — confirm pickup times at booking. Sengokuhara properties (Kinnotake, Hiramatsu, Yoshimatsu) are 25-35 minutes by shuttle or taxi from Yumoto.
Shuttle vs taxi
If your ryokan offers a free shuttle from Hakone-Yumoto, take it — these are scheduled to meet specific Romancecar arrivals and you'll need to coordinate at booking. If you're arriving on a different train or with luggage that won't fit the shuttle, a taxi from Yumoto is roughly ¥3,000-5,000 to Gora and ¥5,000-7,500 to Sengokuhara.
Should you bring the Hakone Free Pass?
For a private-onsen ryokan trip — usually no. The point of these stays is to spend time in the bath and at dinner, not to ride the round-course transit loop. Skip the Free Pass and just buy round-trip Romancecar tickets unless you're planning to do the full ropeway-cable car-cruise circuit on a second day.
Two nights is the right length
One night is enough to do the kaiseki dinner, two long baths, breakfast, and check out — that's already the full ryokan product. Two nights gives you a non-rushed day in between for the Open-Air Museum, Hakone Shrine, or just a long late-morning bath. Three nights starts to feel slow unless you have a specific itinerary planned.