Sangaokayamaryokuchi Park 三ヶ岡山緑地 is a mini hike in Hayama 葉山 that takes roughly an hour - Mount Fuji on top of views of the ocean can be seen on a clear day. It is not difficult but there are lots of steps.
Take Keikyu Bus 京浜急行バス 逗 4 大楠芦名口, 逗 5 横須賀市民病院, 逗 6 長井, or 逗 71 from bus stop ① near the South Exit of Keikyu's Shin-Zushi Station 京急新逗子駅 which is also close to JR Zushi Station 逗子駅.
After 10 minutes, get off at 旧役場前 and walk back where the bus comes from, cross the street, and turn left into a small street with a small vertical sign with only Japanese that says Tsutsuji Course つつじコース.
From there, just follow signs through a residential area then uphill all the way to the top of the 143 m 大峰山 with picnic tables and benches on the way. Eventually, you end up passing the Shinnase Course 真名瀬コース to the coast where people barbecue and hang out when it's warm out.
You can then turn right into a small street at the Shinnase 真名瀬 bus stop towards Kotokuji 光徳寺 and continue to Ajisai Park あじさい公園 before reaching the main street where you can take the bus back to the station at 向原 in front of johnathan's. The way back will cost 200 yen instead of the 250 on the way here.