Is there an airport shuttle or any other service that can get me from SFO airport to the Caltrain Station?

No Direct service

There isn't any shuttle listed. There is a Samtrans bus from the Caltrain station to all terminals. You can take Bart but it is a bit of a chore. They should have put in a walkway or something direct between the people mover that runs to all terminals and the rental car facility but they didn't. Below is from the Caltrain site: San Francisco International Airport (SFO) SamTrans and BART provide service from Caltrain to the San Francisco International Airport. Samtrans Route SFO picks up Caltrain customers on the southbound platform and delivers them to all terminals at SFO (Route SFO is free for Caltrain 2-zone+ monthly pass holders). Customers can also board BART at the Millbrae station and ride to the San Bruno BART Station, where they will need to board another BART train headed to the airport.