If you are willing to walk 2.5 blocks from a bus stop, the Shaw Guest Suite is accessible via transit.
Coming from Lambert Airport: Take Metrolink to the Central West End station (this is a 20-minute train ride). When you get off at the Central West End Station, look for a parking garage to the east. This is where you will find the Central West End Transit Center, which is on the first floor of the parking garage.
At the transit center, look for the 8 Shaw-Cherokee stop, and wait there for the next bus. Here is a link to the schedule (make sure toggle to the correct day of travel and the SOUTHbound direction).
Get off the bus at 39th Street and Magnolia Avenue. If you ask the bus driver, he or she will let you know when you reach the stop. Walk west 2.5 blocks to the Shaw Guest Suite.
Pro Tip #1: For real-time transit info in St. Louis and more than 100 other cities, download Transit App on your smart phone. Authorize a credit or debit card on your Transit App and you'll be able to buy fares in St. Louis. If you are traveling in on Metrolink from Lambert Airport, you will pay the local fare, not the airport fare.
Pro Tip #2: Another excellent resource is the transit feature on Google Maps.
Pro Tip #3: The 70 Grand or the 95 Kingshighway offer much more frequent service. To catch the 95 Kingshighway bus, walk west on Magnolia to Kingshighway. To catch the 70 Grand bus, walk east on Magnolia to Grand.
It's a delightful 20-minute walk in either direction.