Great, thanks for you help so far. I will try to dig out some logs (I think I seen a post on forum how to do it) and copy it here.
This is what I did to upgrade PS
1.backup of everything (copied everything from the root/www catalogue)
2.I put PS under maintenance
3.had one yellow issue with max_size.. sth like that (could overpass)
4.clicked upgrade
5.couple errors poped out in one of the boxes on my upgrade screen (unfortunately, I did not took enough attention to save this somewhere for future reference, my bad)
6.finally it said Finished... (not sure about this part as maybe there was another step and I refreshed tab
7.tried to log back in BO and it stuck on login page
8.logged in to server via ftp and changed dev mode to "true"
9.after that I tried to login again and this time I get this error what I attached to my 1st post.
I have attached some logs.