Are there any tour groups from Shanghai to Mount Putuo? Which one and how much does it cost?

Yes, there are many routes to Putuo Mountain on the Lalagou Travel website. I found the cheapest one to show you. The price is 348 yuan.

Day 1: Shanghai - Mount Putuo

Departure from Nanpu Bridge to Zhoushan via Hangzhou Bay Bridge at around 7:20 in the morning, arrive at Zhujiajian Pier at around 12:30, and then free Activities: Take a small speedboat to Putuo Mountain by yourself, and visit pilgrimage attractions: Zizhulin Scenic Area: Guanyin Yuan, Chaoyin Cave, and Nanhai Guanyin Bronze Statue.

Overnight: Mount Putuo

The next day: Mount Putuo—Shanghai

Continue to visit the pilgrimage attractions: Foding Mountain Huiji Temple, Puji Temple, Dharma Temple Yu Temple, return to Zhujiajian at noon, take the bus to Hangzhou Bay Bridge at around 14:00, see the majestic appearance of the longest cross-sea bridge to date, and arrive in the city at about 20:30; the happy journey ends!