If there is no control panel installed in the server, then it should be a bit tough task to function mysql though we can install it easily.
Sometimes there might be compatibility issues with apace or php.
Why dont you contact any support companies. contact them, they will install/upgrade and do all kind of work which is required to function mysql.
Make sure that you are in the server with root privileges.
Before you proceed with the upgrade run the following command tp know the currently installed mysql packages,
"dpkg -l | grep mysql" (this would list out the currently installesd mysql packages)
Once you get the verion of the installed mysql package you can search in the search engines for compatibilty between the version that is already installed and the one that is availabe in the repository for upgrade.
If you are afraid that upgrading your mysql would break some fuction in the server i recommend that (if possibel )you setup a copy of the server in a virtual environment or another physical server and try upgrading. Now you can check if all is functioning properly. If everything checks out fine you can upgrade in your main server.
Instructions on how to upgrade;
1, make sure that you have your repositories setup correctly
You can find more info regarding this in the followig link,
The only thing here you have to note is that you add the repository links in the "sources.list" according to the version of ubuntu installed in the server.
If you open your current "/etc/apt/sources.list" it would have info regarding the version of the ubuntu. You can also run command "cat/etc/debian_version". You can then search repository for that version in the search engine like yahoo, google etc.
2, Once you have the correct repository info you can run "sudo apt-get update" and you repositories may get updated with some authentication error(authentication key).
To fix this please see following link,
"https://help.ubuntu.com/community/Repositories/Ubuntu". Here go to the title that says "Authentication Tab" . Though this is for graphical interface setting of the repositories you can use the command
"gpg --keyserver [name of keyserver] --recv-keys [keyhash]"
Note here for "[name of keyserver]" is "subkeys.pgp.net"
I think you can use the same but for "[keyhash]" you will have to add the correct key that appeared(the error show whic key its is that needs authentication) after running command "sudo apt-get update".
3, After adding the correct gpg keys for the repositories that you have added run the update command again this time if all keys have been added correctly there would be no key error at the end .
Now your repository is set correctly,
4, Now run .
"apt-cache search mysql"
"apt-cache search mysql-server mysql-client"
This would list out the available package for installation or upgrade for mysql incliding the mysql server and client and also the mysql-php version.
"apt-cache search php5-mysql"
5, To upgrade to available package ,
"sudo apt-get install mysql-server mysql-client"
If the available package in the repository is newer than that of the one installed in the server it would proceed with the upgradation.
It is not that complicated hope you find it likewise too