Originally Posted by Askartus

Just to avoid confusion: The regional train RE 6 is leaving the airport station "Düsseldorf Flughafen" at 13:02 and arriving at Bielefeld at 14:58.
Do not go to the old station inside the terminal called "Düsseldorf Flughafen Terminal" which is only served by suburban trains (S-Bahn)!

And: You may use the 9 Euro ticket for this relation, but then it is 2nd class for you (which, since it is a fairly new RRX trainset, has decent seating). Please be reminded, that it is not possible to buy an upgrade for the 1st class while using the 9 Euro ticket!
A 1st class ticket will be about 60 Euros. And if you want to use an IC / ICE train, you must buy a ticket anyway.

All tickets must be bought before boarding the train, there are multiple ticket machines at the airport station.