Ubuntu does not provide an apt-get repository package for MySQL 5.5 on this older OS, however this is still a widely used long term support version. The following steps will upgrade an existing MySQL 5.1 apt-get version to a standard MySQL 5.5 binary.

Step 1. Remove existing MySQL 5.1 retaining data and configuration

sudo su -
service mysql stop
cp -r /etc/mysql /etc/mysql.51
cp -r /var/lib/mysql /var/lib/mysql.51
which mysqld
dpkg -P mysql-server mysql-server-5.1 mysql-server-core-5.1
which mysqld
which mysql
dpkg -P mysql-client-5.1 mysql-client-core-5.1
which mysql
dpkg -P libdbd-mysql-perl libmysqlclient16 mysql-common
# This will not remove /etc/mysql if any other files are in the directory
dpkg -l | grep mysql
[ -d /etc/mysql ] && mv /etc/mysql /etc/mysql.uninstall
cp -r /etc/mysql.51 /etc/mysql

2. Prepare configuration and required directories.

sudo su -
MYCNF="/etc/mysql/my.cnf"
grep basedir ${MYCNF}
sed -ie "s/^basedir.*$/basedir=/opt/mysql/" ${MYCNF}
grep basedir ${MYCNF}
sed -ie "/^[mysqld_safe]/a
skip-syslog" ${MYCNF}
chown -R mysql:mysql /var/lib/mysql
mkdir -p /var/run/mysqld
sudo chown mysql:mysql /var/run/mysqld

Install MySQL 5.5

sudo su -
mkdir -p /opt
cd /opt
# Install MySQL 5.5 Binaries
apt-get install -y libaio-dev  # New 5.5 dependency
wget http://cdn.mysql.com/Downloads/MySQL-5.5/mysql-5.5.28-linux2.6-x86_64.tar.gz
tar xvfz mysql*.tar.gz
ln -s mysql-5.5.28-linux2.6-x86_64 /opt/mysql
echo "export MYSQL_HOME=/opt/mysql
export PATH=$MYSQL_HOME/bin:$PATH" > /etc/profile.d/mysql.sh
chmod +x /etc/profile.d/mysql.sh
. /etc/profile.d/mysql.sh
echo $MYSQL_HOME
echo $PATH
which mysql

4. Upgrade and verify MySQL Instance

su - mysql
cd $MYSQL_HOME
bin/mysqld_safe --skip-syslog &
tail /var/log/mysql/error.log
# There will be some expected ERRORS in error log
bin/mysql_upgrade -uroot
bin/mysqladmin -uroot  shutdown
bin/mysqld_safe --skip-syslog &
tail -100 /var/log/mysql/error.log
mysql -uroot -e "SELECT VERSION();"
bin/mysqladmin -uroot  shutdown
exit

5. Setup MySQL for system use

# As Root
sudo su -
INIT="/etc/init.d/mysqld"
cp /opt/mysql/support-files/mysql.server ${INIT}
sed -ie "s/^basedir=$/basedir=/opt/mysql/;s/^datadir=$/datadir=/var/lib/mysql/" ${INIT}
${INIT} start
mysql -uroot -e "SELECT VERSION();"
${INIT} stop