Is there a night train or bus Saturday/Sunday from Rutesheim to Flughafen Karlsruhe/Baden-Baden?

As Jörg and Sebastian already pointed out, the railway line between Rastatt (2nd closest station to FKB airport) and Baden-Baden (closest station) is disrupted until further notice, at least until 7 Oct 2017. There's a rail replacement bus service if you need to get to Baden-Baden. The journey planner at www.bahn.de takes this into account.

If you go by train during the disruption, you might prefer taking a taxi from Rastatt rail station to the airport (about 40 EUR). This would save you some time and another interchange.

If you continue to Baden-Baden rail station, you're closer to the airport and the taxi fare will be lower (about 30 EUR, prices are estimates by www.taxi-rechner.de). You might want to pre-book a taxi with one of the companies mentioned on the airport website and agree on a fixed price.

Of course, considering the taxi fares, it could be worth looking for a different flight from Stuttgart ...

A solution using public transport seems really a serious problem at this time of day and with the circumstances described by DCTLib.

That's why I'd recommend a one-way car rental. Using a bigger car rental company like Europcar you could rent a car the day before in Stuttgart (airport for example) and return it at FKB. This should be around 90-100 Euros. You could park it overnight in Rutesheim and then just continue easily in the morning. Should be hassle-free.