Here is short look for upgrading your MySql from 5.1 to 5.5 in a cPanel server.
1. Take the backup of all the databases on the server. Please refer How to create MySQL dump/backup of all the databases
2. Replace the MySQL version from 5.1 with 5.5 in /var/cpanel/cpanel.config file
#vi /var/cpanel/cpanel.config
and then, change 
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"mysql-version=5.1" to "mysql-version=5.5".
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3. Then, run
#/scripts/check_cpanel_rpms --fix
4. After executing this command, try
#mysql -V
MySQL will be upgraded to 5.5. Then try, php -v. You will get the following error:
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root@server [~]# php -v
php: error while loading shared libraries: libmysqlclient.so.18: cannot open shared object file: No such file or directory
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Do an easyapache to fix this.
#/scripts/easyapache
Done!