Since Drush does not run as a daemon, it is very easy to install it in your shared hosting account.
First of all, uninstall Drush 7 if you have it installed. Removing the .drush folder is enough.
Then download latest stable release using the code below:
wget http://files.drush.org/drush.phar;
Test your install:
php drush.phar core-status;
Make it executable.
chmod +x drush.phar;
Rename to 'drush' instead of 'php drush.phar':
mv drush.phar drush;
Since you don’t have root access to your file system, you have to run Drush by providing the full path like this:
/home/your_user/drush do_stuff;
To avoid this, you can store an alias of this command in your .bashrc file:
alias drush='/home/your_user/drush';
Reload your profile:
source ~/.bashrc;
Test again to see if it works:
drush core-status;
Running drush init to enrich the bash startup file with completion and aliases, created problems for me the next time I tried to ssh into my account. If you do that and you have problems, just delete the .drush folder in your home directory.