On the step of the upgrade when I open update.php and start DB upgrading, the porblem occurs:
"The specified file temporary://fileAKOs2U could not be copied, because the destination directory is not properly configured. This may be caused by a problem with file or directory permissions. More information is available in the system log."
I googled for some time and, according to engoogled information, I switched global safe_mode of PHP to "Off" (local, I mean for that virtual host account it was already off). That didn't help.
In fact, problems started a bit earlier, the first problem was that max_execution_time and max_input_time (or smthg like that) of PHP was 60 and it seemed to be not enough for the update.php script to successfully check all of the requirements. I changed both to 180 and it worked.
Then I decided to start upgrade inspite of that the first problem I mentioned appeared. 49% of upgraded was successfull, but every operation (same as initial host.com/update.php opening) was EXTREMELY long, and going to 49% has taken something near 2 hours and then again was this error:
"Fatal error: Maximum execution time of 180 seconds exceeded in /var/www/vhosts/host.com/httpdocs/includes/file.inc on line 1980"