There are a lot of places to see like Mainz, Burg, Bacharach etc....
What can I see in one day.
Biking will indeed be time consuming. So will cruising. You will see more by sticking with the trains, the quickest way to move from place to place. If your goal is to see some small, old-world towns, some castles and some nice scenery, the trains will take do that. The river scenery from the train windows is wonderful, so be sure to sit on the RIGHT side of the train for the best views. You will get in lots of steps on this day... wear comfortable shoes. Get an early start and make BACHARACH your first destination. There's a DIRECT regional express train from Frankfurt (6:59 - 8:03) which will allow you some time to explore the half-timbered buildings there.
From Bacharach, continue north along the west bank of the Rhine on the same train line. Oberwesel (5 min. away,) St. Goar (another 5 min.,) and Boppard (another 10 minutes) are possible destinations for stopovers. Focus on the line just south of Koblenz:
https://www.duesseldorf-blog.de/wp-content/uploads/2008/12/karte.gif
Continue your train hopping to Koblenz (15 train minutes from Boppard,) a rail hub where you can change trains for a southbound excursion along the east side of the Rhine. Braubach and Rüdesheim are probably the two most productive stopover options on this side:
https://rheingaulinie.de/files/rheingaulinie/files/Fahrplan/fahrplan.png
(The actual train time for Koblenz > Braubach > Rüdesheim is one hour, not including stopover time.)
Rüdesheim marks the southern end of the scenic Rhine Gorge section. It's a nice town that in the warmer months will still be active in the evenings. From Rüdesheim it will take 70 minues for the return train to Frankfurt, with trains running late into the evening.
You will not have time for ALL the towns I've suggested stopping in. Do some research on the suggested stopover towns to see what interests you most. St Goar has Rheinfels Castle Ruins. Boppard has a great chairlift ride with a superb view of the river valley. Braubach has Marksburg Castle. Also research the train schedules and put together your own working itinerary, based on your priorities. You can change it as you go, but it's good to have a plan:
https://int.bahn.de/en/
I have deliberately left out Mainz and Koblenz for stopovers, not because they are worthless, but because they are larger places that don't have much to do with the old-world Rhine vibe most people want, and because of the time they would require. You mentioned "BURG"... which means "castle" but is not the name of any town on the Rhine.