Typically, when you download an APK file to your phone, you'll receive a notification that the file was downloaded and you can then "Open" the file from within the notification message. The default action when you "Open" an APK file is to install it.
If you can get a file system explorer app installed such as ES File Explorer, you can do a search for apk files to find the one you're looking for. Do you have a Google account that you can use to access Google Play?