And the airport is the western terminus of it. The stop for the Arch is Laclede's Landing (the last stop on the Missouri side.)
I had forgotten the issue of sporting events; good point. This town is sports mad, and the football, hockey and baseball stadiums are all downtown; I would not try this trip if there is any pro game going on. Any SkyCap could probably tell you if it's a game day, not to mention the Post-Dispatch, which you can buy at the airport for a couple of quarters. If there is no game it is quite do-able in 3 hours with time to spare, though I wouldn't try to take the tram to the top of the Arch; the tickets for that are timed, and you might have a long wait. The ride from Lambert to the Landing has never taken DH more than 40 minutes on an ordinary day. (He does it when he travels for business, his office is at the Landing. Driving tends to be slower.)
The mall mentioned above is Northwest Plaza, at the corner of Lindbergh Blvd. and St. Charles Rock Rd. This center actually does have cabs outside sometimes, but I'd ask the mall concierge to call one for you to get back, to be sure.
It's an ordinary mall, with a large food court and arcade area; the anchor stores are Sears, Penneys, Dillards, and our local branch of the May Co., called Famous-Barr.