To fix this, log into your Synology via SSH (or Telnet), and you need to log on as root (NOT your admin or your user account). The password should be the same as your administrator account.

Oh, and there’s a couple prerequisites here of course. First, you have to enable SSH through your DiskStation if you haven’t already. If you’re on Windows, you also need to get yourself an SSH client like PuTTY. You can also use Telnet if you’re sure about not getting sniffed on your network. 🙂

Mac: use the terminal app

Log in as root: ssh root@192.168.1.xxx (your disk stations ip) and enter the admin user’s password,

And enter the following command:

chown -R nobody:nobody /volume1/web/wordpress

Substitute your WordPress folder for ‘wordpress’ above. Now try to update and you should be up running again.

If instead all you got was a big stream of ‘Operation not permitted’ messages, it’s probably because you didn’t log in explicitly as the ‘root’ user.

Source: http://www.darrenmonahan.com/hardware/synology/updating-wordpress-on-a-synology/

Note for NSM 6 and above, you need to do the following:

admin@192.168.0.xxx

Password:     (Enter admin password)

sudo -i

Password:     (Enter admin password)
root@192.168.0.xxx:~# chown -R nobody:nobody /volume1/web/wordpress   (change destination based oun your config)

See these 2 articles

https://forum.synology.com/enu/viewtopic.php?t=126166

https://forum.synology.com/enu/viewtopic.php?t=116102&start=75