There are four convenient ways to travel from STL to the Hyatt Regency St. Louis at the Arch, each with different costs.
Approximate Cost: $2.50-$4.00
Directions: From Terminal 1 exit using Door 1 (ticketing level, east of all ticket counters), and from Terminal 2 exit using Door 12 (through the parking garage). Take the MetroLink to "Pine". The hotel is three blocks away.
Airport Best Shuttle
Approximate Cost: $16.00 one way; $32.00 round trip
Phone: 314.333.1043
Promo Code: USAS
Approximate Cost: $30.00-$45.00 one way
Directions: Use the Uber and/or Lyft apps to book a ride and proceed to the designated rideshare area at STL for pick-up.
Approximate Cost: $50.00-$75.00
If you are arriving via ground, there are three parking options available. 
Address: On-Site at Hyatt Regency St. Louis at the Arch
Self-Parking: $30/day
Valet: $40/day
Manson Garage
Address: 330 N 4th St. (6' clearance)
Kierner East Garage
Address: 500 Pine St
Note: No in/out privileges