I am traveling from Frankfurt airport to Karlsruhe in Saturday evening. I will reach Frankfurt airport around 18:40. Is there any direct train/bus from Frankfurt airport to Karlsruhe that time? Is there any train in Sunday morning?
Here's an actual link to bahn.de for what you're looking for:
There's a train about once per hour. The trip itself takes just over an hour. Depending on which train you take, you might need to change trains in Mannheim. The "Umst." column shows how many train changes. If it says 0 then the train is direct.
If you do need to change trains, you have 13 minutes to do it in, which is rather a lot, so you should be able to do it with no problem. Here's a diagram of the Mannheim Main Station. It looks like lines 4 and 5 are even on the same platform, so you won't even have to go up or down to switch lines; just go to the line on the other side of the platform (and verify that it's the right train before getting on board!).
Of course, you should verify all this on the day of travel, and in particular, watch out for late trains, which Deutsche Bahn is notorious for. But it should be an easy journey.
Yes, there are trains to Karlsruhe departing from the airport's long-distance station (Fernbahnhof) every hour or so. You'll likely have to transfer to a different train once you get to Mannheim. For exact times and routes, go to bahn.de and search for trains going from "Frankfurt(M) Flughafen Fernbf" to "Karlsruhe Hbf".