Hi @Slava_Knyazev welcome to the Asana community 
When you do make these customizations, I am assuming that you are clicking on “Save view” once you’re done with the customizations?
Also, you’re correct it doesn’t seem to sync to mobile devices. I did some digging around here and it looks like this is something thats not currently possible. I just voted on this as this is something I didn’t realized I needed until I saw this I suggest you cast your vote as well.
From what I see these saved views gets applied in the desktop app and in the web browser. So as an alternative it is possible to get save view on mobile if you use Asana through the web browser on your phone.
To accomplish this unfortunately this will require you do some finagling
When do go to the Asana website or rather your project it will try to get you to download the app but, WAIT
depending on what web browser you’re using on your phone you need to navigate and find how to enable Desktop View from your mobile phone
Most web browsers will provide this functionality it will just look different for you, when you find it, you should now see your saved views
Hope that helps 