Public boat on Rhine

There are day-cruise companies that sell passage to the public as well as ferries that cross the river in certain places.

Cruises:
https://www.k-d.com/en/
http://www.bingen-ruedesheimer.de/timetable/?lang=en

Ferries: Scroll down to the "Mittelrhein" heading to see which towns are served:
http://www.rheintal.de/daten-fakten/verkehr-transport/rheinfaehren-personen-fahrzeuge.html

There is a one-day pass you can buy for the KD boats on the Rhine but I personally don't thinks it's worth the price. They don't run frequently enough to make it worth it to use as transportation to get from one town to another, so I wouldn't say it's like a public bus. Taking a single cruise on one of the boats is a wonderful way to spend a few hours, but for faster and cheaper transportation from town to town using the train and the cross-river ferries makes more sense.

Download the suggested timetables, noting the dates of your trip against the seasonal timetable that applies. The boats are not frequent enough that you are going to see three towns in one day. You need a strategic plan, and the much faster trains, and the knowledge that downstream boats are faster (but not so fast that you miss the view; it's fine downstream.)

Use the search box, with maybe "middle rhine" at top center to read about the most popular towns to visit and the boat ride to take.

If you want to see the towns and the castles from the river, take the river cruise (preferable from Bingen to St. Goar). If you want to tour the towns, use the train.