Pretty much every car service uses the same Mercedes/Peugeot vans that have 3 seats facing another 3 seats in the back. They run about $70-90 USD depending on how you book it, or you can do it on demand for about 100 euro with the Uber app. I'm guessing it can be done cheaper, but those are the ways I've done it. It's worth it after a long flight if there's light traffic - I suppose you could just check Google Maps and decide based on the traffic at the time whether to take a car or the train.

Whether the train will be crowded depends on the time of year, time of day, day of week, etc. But you'll probably have seats, just maybe not together, if its busy. From a long haul flight I'd probably just book the car service, but if you're coming in on a full night's sleep, the train does more or less guarantee you a 30 minute ride.