Anytime cards cannot be played at any moment. They can always be played on your turn in the Rescue and Rare Finds phases. They can also be played before or after specific points in the game. You can change when the game will ask you to play Anytime cards by clicking on the "When to play" button. The button is only displayed on some Anytime cards that can impact other players. You can click on the gear icon in your player panel for more options related to this button.
Like in the real game, you must tell other players that you pass. When passing, you can also play Anytime cards if you wish. If you do not want to be asked to pass, you can click on the gear icon in your player panel and enable the "Auto pass" option.
There can be a lot of information to view at the same time. You can reduce the amount of scrolling you have to do by showing (above your player boat) small shapes that represents cats and treasures. Those small shapes are shown when you have to place a cat or a treasure but you can enable an option so that they are always shown.