Late June is wet and hazy — Fuji hides most days, so treat any sighting as a gift, not a plan. Best odds are early morning before the cloud builds: from the Owakudani ropeway (~15:00 is decent if it's clear) and across Lake Ashi near the torii. The cleanest full-mountain view is often from the shinkansen window (right / Side E) just after Odawara on the 25th. Don't chase it; if the mist stays, that's Hakone being Hakone.
Hakone sits higher than Tokyo — reckon on 18–24°C, noticeably cooler in the evening, with the tail of the rainy season (tsuyu) meaning frequent drizzle and low cloud. Pack a light layer and a fold-up umbrella; the damp is exactly what makes the ajisai glow and the onsen feel right. Indoors at the ryokan you'll live in the yukata.