In your original post you fly out on the 28th. Now it's the 27th in your most recent post.
Either way, with Sprantal and later Hamburg in your sights, you are going to spend 2 of your 4 or 5 free days making extremely LONG ground trips. Going to Munich puts you in transit on yet another LONG journey. And you can't find a car. And even if you do find a car, it's going to cost a fortune, then add gas at $8/gal... So this is no way to spend your time, really, and it's no surprise that your planning is sort of at ground zero - I would be in the same shape with those ideas in mind.
The following is a template for you to work with which includes Sprantal, castles, and the long southbound then northbound journeys that must be made.
7/23: Amsterdam > Karlsruhe, 8:06 > 13:47 by train; pick up car, drive to Bretten/Sprantal for afternoon and evening night.
7/24: Visit Sprantal area in morning; return car to Karlsruhe within 24 hrs.; train from Karlsruhe > Rüdesheim (14:00 > 16:35) for 2 nights there.
Rüdesheim is a gateway town to the castle-studded Rhine Valley Gorge, Germany's most scenic train route and a UNESCO World Heritage site. There's no better place to see/visit castles on the way north to Hamburg, or in all of Germany, really.
7/25: Morning river cruise past vineyards and Rhine Castles north to St. Goarshausen 9:00 > 10:45; catch train north (22 min.) to Braubach for lunch and Marksburg Castle tour.
Braubach's old town (scroll through photos) is a nice for a stroll and lunch. Uphill walk of 25 minutes for the castle tour.
Return to Rüdesheim by direct train in afternoon.
7/26: Check out; train north to COLOGNE for a stopover of several hours... See the cathedral, the NS Doc center, museums, whatever interests you. Catch an afternoon train to Hamburg. Total time on the train this day will be around 6 hours.