You can ask us to install Sphinx on your hosting account.
You probably need to generate the Sphinx config file from a Magento extension.
You should configure Sphinx to run on the IP address of your hosting account. The default is probably localhost / 127.0.0.1 so you will need to change it.
You can use the default port, which is probably 9312, 9314 or 9315.
You can restart Sphinx using this command:
How to use Sphinx depends on the Magento extension you use.
Mirasvit Ultimate Sphinx Search
Miravit Ultimate Sphinx Search is the most popular Sphinx extension for Magento 1.x.
You need to make sure its "Engine" is configured to use the External Search Engine using:
System > Configuration > MIRASVIT EXTENSIONS > Sphinx Search > Configuration
By default it is configured to use MySQL as engine which is slow and uses a lot of CPU.
To be able to start the Sphinx daemon and to run a Full Reindex, you probably need to generate at least one index first using the Admin menu Search > Manage Search Indexes.