There is a bug on Ubuntu 16.04 with systemd-timesyncd. Firstly the problem is because systemd-timesyncd.service has parameter that prevents starting it on OpenVZ (depending on man this should work without problems, but this is not true):
The solution for this part of problem is here: systemd-timesyncd.service inactive on Arch on VMWare. However, this still doesn't resolve problem at all because when trying to start it with ...
sudo systemctl start systemd-timesyncd
... results in something like this:
Aug 01 10:31:02 ingarill systemd: systemd-timesyncd.service: Unit entered failed state. Aug 01 10:31:02 ingarill systemd: systemd-timesyncd.service: Failed with result 'exit-code'. Aug 01 10:31:02 ingarill systemd: systemd-timesyncd.service: Service has no hold-off time, scheduling restart. Aug 01 10:31:02 ingarill systemd: systemd-timesyncd.service: Start request repeated too quickly.
I copied this "error" part from https://www.reddit.com/r/linuxquestions/comments/4vln5o/cant_start_systemdtimesyncdservice/ where similar error is described (I don't think it is connected to this one at all). After that systemd-timesyncd state is inactive and last state is described as "Shutted down".
So that's what I discovered so far, hopefully enough for helping me resolve the problem. Thanks in advance!