Each race you play will cost you one ticket. You will receive a free ticket for the Gold Cup Event every 8 hours but can also buy more tickets in exchange for coins.
If you run out of both tickets and coins, you can complete races in the Daily Cup to win a chest with Treasure Cards. Some Treasure Cards Sets will reward you with coins which you can use to buy tickets for the event! You can also get Treasure Cards by completing Daily Quest and some side missions (with Chest icons over them) that you will see next to the map. Simply tap on the purple book on the right side of your map screen to claim rewards for completed sets.