I would say this it is very unlikely that you can install a game server on your shared web hosting account. You would over exceed the allocated resources to run that and your account would likely get terminated.
Yes you need at least a VPS or your own Dedicated Server in order for this. I don't think any web host out there lets you utilize the amount of CPU usage that a counter-strike 1.6 server does on a $4.99 shared host plan, its unreal. Also not to mention you need SSH access in order to run the commands.
You need to get a VPS or Dedicated Sevrer, also need the game files.
you will have to buy a game server, a dedicated server or at least some VPS (virtual private server). Thing is, shared webhosting does not allow you to execute any files and such, its only used for a website.
I know if you want a source game (linux) go for 512mb ram (guaranteed) and a single or 2 core and 2.0ghz at least. That will MINIMUM cost you 20-30 dollars and sometimes it costs more than 50-60 dollars.
I suggest you go for a game server provider, they are usually between 1 and 2 dollar each slot, which is way cheaper, AND they usually provide unmetered bandwidth, so players can download all maps they want, and unlimited map/mod space.