Unfortunately, no App store on your Apple TV means that you do not have a 4th generation or higher Apple TV. Currently Fullscreen is only available to download on 4th generation or higher Apple TVs. However, you can use Airplay on 1st, 2nd, and 3rd generation Apple TVs.
For Airplay, make sure your iOS device and AppleTV are on the same Wi-Fi network. Then when you start playing a video, tap the Airplay icon in the bottom right hand corner and select the AppleTV you want to stream to. Use the controls on your phone to play/pause and fast forward/rewind.