Just open your browser, find the APK file you want to download, and tap it – you should then be able to see it downloading on the top bar of your device. Once it's downloaded, open Downloads, tap on the APK file and tap Yes when prompted. The app will begin installing on your device. Simple.

Just make sure to have the Uknown services enabled. Inside security.

What are unknown services?
This will allow installation of apps from unknown service on the internet. If you have that disabled Android won't allow you to install unknown service apps that are outside of the market store.

When it comes to new versions you will have to manually download a new one!

If you live in USA Canada or Japan just search Fate Grand Order and it will work. However, if you come from any other countries you will be greeted by this nasty image!

A free VPN with a connection to Japan or USA. (Keep in mind if you register Fate Grand Order in Japan it will not work to log in to USA version of the game).
For this tutorial I used (Opera Free VPN): https://play.google.com/store/apps/details?id=com.opera.vpn&hl=en

1. First Download your Free VPN or Paid VPN if you already have.
   
2. Connect to the USA or Japanese Server.
   
3. Exit Google Play Store.
   
4. Go to Settings find Apps or App Manager and inside find Google Play Store
   
5. Click Force Stop then Storage.
   
6. Inside Storage Clear the data and cache.

If you have your billing address set to your country or have PayPal or credit card set to your country this will not work! Make sure to set the billing address to USA or Japan and remove your card. Because Google Play Store will detect your location and set it as your country. Once your billing address is set to USA or Japan or is removed you are good to go!

Google Play Store doesn't need your billing information or card so remove it all together.

You can open Google play store and search for Fate Grand Order and click install.

Proof that it worked for me I come from a Sweden which is outside of USA, Canada or Japan.


FYI
Make sure to have VPN always open force stop your google play store and clear history. If you have a Google account connected to a country other than America make sure to make a new account with an American address this is needed otherwise it won't work!

To update using the APK method just delete the old APK file and replace with new.
The update using Google Play Store method:
This is done automatically you will not need the VPN.
Once you finished the Google play store method guide you can delete the VPN and your region will be set back to your country and you can change your billing address back to default. This will not affect the game updating.

Congrats and enjoy your game!