I was sitting at the George Sherman Union the other day when an older gentleman approached me and asked about the sticker of actress Lauren Bacall pasted on the back of my laptop. After bonding over our shared love of Jean Arthur comedies from the 1930s, the man posed a question: How can we get more kids your age to watch old movies?

It seems as if every figurehead of the film industry has tackled “old Hollywood” over the last few years. From the jaunty Woody Allen production of “Café Society” to the Oscar-winning “The...