After Effects preview with decreased image quality I have some 3-4 year old AE projects which use bigger png textures. When I created, them they looked amazing in the preview. The quality was render quality. The images were displayed as they look like when I open the source png files. 

So the problem is that when I open them now (with the exact same computer, with a newer AE version), the images look terrible with the preview. The quality of the textures has been lowered a lot. Changing the preview resolution to "Full" doesn't help. I am unable to preview my composition with normal quality while I can render it in almost real time, so it's not because the file ate all of my RAMs. 

I would really appreciate if somebody could give me some tips about how could I try to solve the issue.
You're probably using "Fast Draft" as your preview. This is a new preview mode which uses the GPU to provide "fast"* rendering for previews. This mode will generally look pretty bad, with less or no antialising, lower resolution textures, and less accurate colour.

You can go back to the old adaptive resolution rendering for previews, or use final quality (it actually works faster sometimes, go figure) using the little widget at the bottom of the comp window:



*Adobe's words, not mine, IMHO the latest versions of AE are the slowest yet