Frankfurt to Dusseldorf

http://www.bahn.de/i/view/DEU/en/index.shtml check here for trains, put in Frankfurt/Flughafen Fernbf (long distance train station) and Duesseldorf, don't forget the e after the u, right now there are still savers fares for as low as 29,-- € available, but with those you have to take the booked train or loose your ticket, walk up fare is 80,-- €, you can buy that after arrival.

There are 2 train stations at FRA connected to terminal 1, downstairs is the regional one and upstairs the long distance one, that is the one you need.

https://www.flixbus.de/ can beat that price but doesn't have many connections. Also, it's slower than the train, more than twice the duration that the train (which is direct in some cases) offers.