You can still get to Kassel at that time by train, e.g. with Deutsche Bahn, tomorrow, Wednesday, departure 19:02 at Frankfurt airport station ("ICE Fernbahnhof"): http://reiseauskunft.bahn.de/bin/query.exe/en?revia=yes&existOptimizePrice=1&country=DEU&dbkanal_007=L01_S01_D001_KIN0001_qf-bahn_LZ003&S=Frankfurt+Flughafen&REQ0JourneyStopsSID=&REQ0JourneyStopsS0A=7&Z=Kassel+Hbf&REQ0JourneyStopsZID=&REQ0JourneyStopsZ0A=7&trip-type=single&date=We%2C+17.12.14&time=18%3A55×el=depart&returnTimesel=depart&optimize=0&travelProfile=-1&adult-number=1&children-number=0&infant-number=0&tariffTravellerType.1=E&tariffTravellerReductionClass.1=0&tariffTravellerAge.1=&qf-trav-bday-1=&tariffTravellerReductionClass.2=0&tariffTravellerReductionClass.3=0&tariffTravellerReductionClass.4=0&tariffTravellerReductionClass.5=0&tariffClass=2&start=1&qf.bahn.button.suchen= Savings fare is 19 EUR only, regular fare is 59 EUR. There's a limited amount of savings fare tickets, make sure check for your actual day of arrival, and book in advance (credit card). Also, if 18:55 is the arrival of the plane, you need to plan in at least 30 minutes for baggage reclaim and customs, maybe more if the plane gets delayed etc. Alternatively, take a "Fernbus"; there are many companies recently, prices are low so some companies are struggling. Use a search engine to compare offers. There are cheap hostels (very simple rooms with multiple beds, used mostly by young travellers) in Frankfurt, use "Frankfurt hostels" in a search engine. Prices start at around 10 EUR/night, e.g. http://www.aohostels.com/de/frankfurt/frankfurt-galluswarte/?Seed=HostelFrankfurt&gclid=CJvWjunMysICFdTLtAodyUwAKg http://de.hostelbookers.com/hostels/deutschland/frankfurt/ and many more.

You have international driving license? Then just rent a car. Or check bahn.de and buy a ticket in advance. Visa is fine. Don't make it tight, delays happen, and the passport lines move slowly.....