Japanese movie theaters are top notch, and English speakers shouldn't write them off. The Toho Cinemas chain always has at least one English film, often a blockbuster that debuted in the US many weeks before. Those are fine, but the real draw at the Roppongi Hills location is the 10 am classic. Recent showings were Stand By Me from 1986 and Bonnie and Clyde from 1967.
You can reserve a seat online beforehand or at one of the kiosks when you arrive. The Stand By Me screening I attended had only two open seats when I arrived a half hour in advance. On a Sunday morning! Warmed my heart as a lover of that movie.
The classics are shown in the Premier theater, which has reclining chairs, blankets, and tables for your snacks. Premier tickets cost $30 in the evenings but only $10 for these morning screenings.
The tradeoff for expensive tickets is that snacks are way tastier and cheaper than US theaters. Caramel corn, anyone?