Is there some way to get the new APK from Google Play Store sooner? For example, Pokemon Go 0.57.2 was available on apkmirror.com for 4, 5 hours now, but still it is not on Google Play.

Is there any way to get it from official source sooner?  I mostly trust apkmirror.com, but say, what if somebody hacked into apkmirror (which I assume will be less secure than Google Play Store) and target a country in the world and serve a modified apk, or target some IP, then it will be not secure.
If you are taking about 



  Pokemon go


in specific. Just for your information, app is still available in playstore. 
Don't know from which country you are in. I'm in India. And I can find Pokemon Go in playstore 

Below Link
Pokemon Go Playstore 


If you talking about any playstore apps, I don't think there is any way you can download it apart from source(playstore).
If any website found which allows to download apks . I would warn you not download it and install. Those might be injected and can cause harm to ur details and your contents in your devices.


Note: there are some apps in Google Playstore that can slow down and bring harm.