Saudi Arabia wants to do more than export oil. So it’s importing popcorn machines.

On Wednesday, when moviegoers in Riyadh take their seats to watch a screening of Walt Disney Co.’s “Black Panther,” it will be the first time a Hollywood movie has played in a theater in Saudi Arabia in 35 years. AMC Entertainment Holdings Inc., the world’s largest movie theater company, is the first to operate in Saudi Arabia since a longtime ban on cinemas was lifted five months ago.

AMC’s...