The only way to login to your account on another device, or after deleting your app, is to login with Game Center or Play Services. Please be signed in with your Game Center/Play Services account, and then start Fun Run Arena. Since there is no other way of signing in to the game, it is very important to connect your Game Center/Play Services account to your Fun Run Arena account. This is done the first time you start the app or in the Settings screen.
I still can't login to my old account!