If you have a ticket for a German regional train you can use it for all regional trains (not ICE or EC/IC) the same day. If you are in Aachen or Köln, there are many trains going to Köln or Düsseldorf, so if you miss one the next will come soon. And as others have said, no reservation on regional trains.
If you want the cheapest connection:
Look at the site of DB. https://www.bahn.de/p/view/index.shtml switch it to English (up in the first line left of the center is written Deutsch, clicking on it gives you a choice of languages. Fill out start, destination and the date/time fields. Below these lines untick the field prefer fast connections. Go down to the grey field further options on the new page opening go to Connections > more means of transport Click on more means of transport. Then some lines appear, untick the box ICE (this excludes all Thalys and ICE trains). Then you get cheaper trains, Belgian IC and German regional trains, but with an absurd in-between connection between Welkenraedt (B) and Aachen (D). This is the cheapest connection AFAIK. Takes 3:30 hours. If this is too long for you, take ICE or Thalys.