MICHAEL ARENELLA is at an open-air flea market in Lambertville, N.J., his fedora-topped head leaning over a gramophone. He winds the crank, lifts the arm and places the needle on a 78. Out of the old oak box comes a voice from the past: Arthur Fields singing “In My Tippy Canoe” in 1921. Mr. Arenella stands silently, hands in his pockets, taking in every crackly note.

Other shoppers stop to figure out where the music is coming from. To their delighted surprise, they see Mr. Arenella, a 34-year-old jazz musician and bandleader from Brooklyn who looks as if he had stepped through some wormhole in the space-time continuum. He is 6-foot-1 and dressed in windowpane-checked pants, a blue paisley ascot, a red-and-white checked shirt, a herringbone vest, a blazer with a blue pocket handkerchief, cap-toe faux-crocodile ankle boots, a pinkie ring and a brown fedora. A few people wander over to check him — and the gramophone — out. But most just stare, smile and walk on, this vision of the ’20s brightening a mundane, modern New Jersey day.

“I need to buy this,” Mr. Arenella says, laying down $375 in cash for the gramophone, making it the third in his collection....