Yes, it's a completely touristy thing to do, but you can't travel up into an arch anywhere else in the world, so why not? Winter hours are 9a to 6p daily. Summer hours (Memorial Day to Labor Day) are 8a to 10p. Trams to the top of the arch leave every 10 minutes and the tour lasts for about an hour. The last tram leaves 1 hour before closing.
You can purchase tickets online to avoid any potential crowds at the box office. There are a variety of "combos" to choose from. Tram only pricing is $12 for adults and $8 for kids (3-15).