Sadly, it's a seasonal event that only occurs once every year around the same time (the Rising celebrates FFXIV ARR's launch and metaphorical rebirth from the disaster that was 1.0), and each year features a new quest, with the previous years' events no longer being accessible (SE actually mentions this in the event info page for every seasonal event on the Lodestone). Item rewards from seasonal events are often added to the Mogstation for purchase much later though, so the 'Cheep Dungeon' minigame and the items the event rewarded may probably be available in the future.

But sadly the event is gone for ever I'm afraid.