It's a very steep walk up to the Bacharach hostel or a taxi ride. What time do you arrive in Bingen/Bacharach? If you head straight up to the castle once you're in bacharach, you are probably not going to want to walk back down and then back up again that same day, which sort of eliminates sightseeing options. And you have a bunch of those on your plate.
I suggest you take the boat from Bingen straight to St. Goar. Bingen-St. Goar is the most scenic part - continuing to Boppard isn't obligatory and may be a little repetitive anyway. Can you catch that 14:30 boat in Bingen? That would give you time to drop bags at the St. Goar TI office (free storage service) and see the castle. Return to the TI before 6 pm for bags.
If you are staying in Bacharach you can catch a train there (10 minutes) then taxi or walk up. Or you could just stay both nights in St. Goar. That's what I'd do.
The following day you can leave your bags in your St. Goar hotel room (if you stayed there the first night) to explore the other towns (Bacharach, Oberwesel?) OR... if you stay in Bacharach you could pack up, check out, head down into town and catch a train to St. Goar to drop bags at a hotel there (but it would maybe be a bit of a shame not to explore Bacharach a little, and how do you do that with bags?)
And that afternoon, you can cross the river in St. Goar (ferry) and take the train to Braubach for Marksburg (about 22 minutes from St.Goarshausen.) The 4 pm tour is in English.
I know, staying in a castle sounds romantic, but in fact the interior is sectioned off into dozens of modern hostel rooms. And this all works more conveniently from an in-town base, preferably St. Goar, for 2 nights.