In the hourly model your server is only charged while RUNNING. Stop your server whenever you like — billing stops the moment it does. If your balance runs out, the server is put to sleep automatically.
When a server enters sleep mode, its world, plugins and configuration are archived; when you restart it, they are restored and everything is as you left it. Without the File Protection add-on this archive is kept free for 14 days — if you don't start the server or top up enough within that time, the world is reset and the server starts fresh (the server itself is not deleted, only its world is renewed). With File Protection your world is backed up daily and kept indefinitely (pay as you go: ~₺2.50 per stored GB per month).
Once you top up your wallet, you wake the server with one click; everything is as you left it.
You can also stop the server yourself at any time — not a single kuruş is charged while it's off. This is the most economical pattern for groups that play occasionally.