Auf Schönburg is an amazing place and a good choice. Just one night there?
I'm with Jo. The train is best - and there are some DIRECT trains to Oberwesel from FRA. Auf Schönburg, though, will require a car (or taxi from the train station.) Your transatlantic flight (from SFO??) means some level of sleep deprivation and an inner clock that is way off... I think by doing an extensive amount of driving and sightseeing on your first day that you are undertaking far more than is wise for most people. Your flight may or may not arrive on time. Luggage retrieval, passport control and customs typically take a while at FRA. The cruise from Bingen to Boppard takes 2.5 hours - and before that you have to drive to Boppard, then take the train south to Bingen, then get to the dock. Then you're going to drive to Eltz, take a 1-hour walk-through tour, and drive back to Oberwesel... What time is it now, anyway?? I've never driven drunk, but I have driven groggy, and I tend to equate the two experiences... Knowing how I normally feel after crossing 9 time zones on a red-eye and spending 12-15 hours in planes and airports, I absolutely wouldn't try to squeeze all that in one day and drive nearly 100 miles too.
If I had to drive, I'd load up on caffeine at the airport and head straight to Oberwesel and drop bags at the castle-hotel front desk. Then I'd drive down into town, look around, grab a bite, walk the old town wall (really nice walk) and see how I felt. If I felt like a cruise, I'd take the train to Bingen - there are about 5 different trains between 11:30 and 1:30 - and catch the 14:30 boat north. Maybe I could stay awake for all the castles. I'd probably get off in St. Goar at 15:55 - I will have seen the most intense part of the river already - and then maybe visit Rheinfels Castle there before returning to Oberwesel by train and driving back up to enjoy the amazing views from Schönburg for the rest of the day (and an evening meal there.)
The train fare to Bingen is 4.20€ - but show your ticket at the KD dock, and you'll get a cruise fare discount for about the same amount.
Burg Eltz: I'd see it the next day. You can see the Rhine castles north of St. Goar on the way to Eltz by car or by train.