Ok, I am not familiar with the mate configuration, but I should have an iso lying around on my other machine. When I get hold of it (my wife is currently shopping online) I can start it in a vm and see how you can configure that back.
In the meantime, you could look in the menu under categories like settings or system and see if you can find the update manager there.
Also, if you are feeling bold, you may want to give it a try and update on the command line. It's really not that difficult.
All you need to do is opening a terminal and running
sudo apt-get update
This will basically look for available updates.
Once that's done run
sudo apt-get dist-upgrade
This is the actual system update. If you are asked for a password type your user password.
It may take a while, make sure your system is connected to power and do not interrupt the process once it is started...