You can walk the grounds and browse the museum and visitor center without a ticket. A timed ticket is needed only for the tram ride to the top.
Tram ride to the top: A timed ticket is required for the tram to the top of the Arch (adults from about $15, prices vary by date). Buy at gatewayarch.com, by phone at (877) 982-1410 or at the park, and book ahead on busy days.