Fly to Stuttgart (STR) airport and take a cab (about 50 Euro) or rent a car. The airport is located at about 40 km from the institute. There is also a direct bus to Tübingen (RBS Linie 828). It takes about 1 hour and you should get off at "Pauline-Krone-Heim". This stop is close to some of the hotels and you can walk uphill (10 Minutes) to the institute or take another bus (Line 2) from the stop "Linsenbergstraße" across the street and get off at the next stop.
You can also fly to Frankfurt (FRA) airport and follow the instructions "by train" below (approximately 2.5 hours to the institute).