jackal

It's a 30-minute MetroLink ride. Add in a couple of minutes of walking on either end and you'll probably want to leave the airport terminal no later than 3pm to arrive at the Amtrak station around 3:40pm at the latest (20 minutes to get on the train is plenty, assuming you already have your Amtrak ticket on your phone or printed out). Earlier would of course be more comfortable; trains run every 20 minutes. Were I in your shoes, I'd probably aim for the 2:27p train for comfort, meaning I'd want a flight that is scheduled to arrive no later than 2p without luggage or 1:45p with luggage. That gives me room for up to a ~45-minute delay and still easily make the train.

In a pinch, you can cab/Uber it from STL directly to Kirkwood (~30m), which would give you an extra half an hour (scheduled departure from KWD is 4:29p).