Assuming that iCloud is set up correctly (as shown here), your cards should automatically update on all your devices, usually within seconds (there is no “sync” button that you have to press).
However, sometimes it can take a while, from minutes to hours, for your changes to appear on your other devices. Usually this is because Apple’s iCloud servers are at their capacity and uploads to iCloud slow down. Typically this problem goes away on its own after a while as the iCloud catches up with the demand.
3rd part app developers have very little control over iCloud. If iCloud is slow, usually it is not the fault of the app (the Flashcard Hero decks are stored in a way so that only the changes made to a deck have to be uploaded to iCloud).
Things you can try to speed up iCloud:
By having the iPhone and Mac access the same Wi-Fi network, iCloud can transfer data directly between the devices instead of taking the “detour” through the iCloud servers. Eventually the data will be uploaded to the cloud, so you’ll still need a working Internet connection.