Moved out of Stuttgart in 2010, so experience a bit dated, but I think it's unlikely the road-traffic situation has improved at all.

Airport is in the very south, AMG up north - which means either taking a bigger detour to avoid the city and rushhour ploughing into it - or face that two ways.

S-Bahn airport/Kirchberg is 1h15m, Kirchberg/mainstation 45m. ICE (express to Munich) is 2h15m leaving hourly (15:19 one should fit your goal).
Kirchberg is outpost-ish so dont expect to walk up to a waiting taxi, better prearrange by Uber or the aforementioned myTaxi-app -- or ask the host to pick you up in Kirchberg or Backnang?

HTH and have a safe trip