I've edited your question to add the details from your comments. If your question is which train you will be able to catch, it's important to know exactly when you're landing (20:45, not 21:00 as originally written), which terminal you will land and if you have checked luggage or not.
German Railways gives you an estimate of 10 minutes to walk from Terminal 1 or 25 minutes to walk from Terminal 2 to the railway station. These estimates are usually excessive, so if you walk at a normal pace and are able to follow the directions without problems, you could plan with less time.
If you are scheduled to land at Terminal 1 with a domestic or intra-schengen flight (no immigration required) at 20:45 and have no checked luggage, you should have plenty of time to get to the railway station and buy a ticket before catching the 21:53 train to Karlsruhe, arriving at 23:00. From December 14th, the train departs one minute later, at 21:54. If your flight should be delayed, there are further trains departing 23:17 and 00:28.
The regular price for the train ticket, which you will have to pay if buying the ticket shortly before departure, is 40€. If you book your ticket well in advance, there are saving offers starting at 19€, but these are bound to a specific train. If your plane is delayed and you have to take a later train, the rebated ticket is forfeit and you would have to buy an additional full-price ticket for the next train. It's up to you to take this risk or not.
If the 21:53 train is not fast enough, the only realistic option to be in Karlsruhe before 23:00 is IMHO to take a taxi. The estimated price is about 220€, but with not much traffic so late in the evening, the trip shouldn't take much more than an hour and with some luck, you would be in Karlsruhe already around 22:00.
If you consider the time you need to pickup and return a rental car, it will take you longer than going by train and at least when including fuel, you are not likely to find a rental car for less than 40€.
Bus or finding a private driver (Mitfahrgelegenheit) willing to bring you are both probably the cheapest options, but if time is an issue, they are both less than ideal.