To enter and play the Stella's Lucky Elevator event, you need an Elevator Ticket. You can sometimes collect them as rewards in different Events around the game, or you can purchase them in the in-game Shop while Stella's Lucky Elevator is live. These Tickets DO NOT expire, so you can save them up for a later date if you wish.
When you use an Elevator Ticket to join this event, you will start at elevator shaft 1 (out of 50 total).
If Luna catches you, you can use another Elevator Ticket to continue playing without losing your rewards (unless she catches you again)! The number of Tickets needed to proceed within the event will increase with every time Luna catches you, up to a maximum of 10 Elevator Tickets per continuation. If you finish the event or exit the event early, the number of Elevator Tickets needed to continue will reset.
See also: What are Express Tickets?