You're likely describing Shatranj — the Persian/Arabic predecessor to chess (7th–15th century), or a pedagogical variant.
Historical boards rarely had arrows. Teaching diagrams in manuscripts (Libro de los juegos, 1283; Persian treatises) used arrows to show piece movement. Some modern travel/educational sets print arrows on squares for beginners.
Share: material (wood, bone, plastic), box art, any text/languages, piece shapes (figurative vs. abstract), board markings beyond arrows.
Shatranj is the best fit for "old simplified chess with knights/kings." Arrows = teaching aid, not standard equipment.