For prolonged freshness, store the apples in the crisper drawer of the refrigerator. They will usually remain fresh for at least one month.
Apples continue to ripen after harvesting by releasing ethylene (a gas). Refrigeration will slow the release and ripening. You would still find trace amounts of ethylene being produced, which ripens other apples stored nearby.
Hence, store them in separate crates/trays, far from one another or any other fruit, for best results.