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Ultimately, you are not limited by using a shared host as long as you can use a scripting language.
Create a script that will perform the task you want to perform when you access a web page. If there is no output from the task, just have the page do the task and produce "Success". (Or an error message, etc.)
Then, on a Linux or Windows system at home, use cron or a Windows scheduler to hit the URL periodically.
Oh, you will probably want to password-protect the page.
If you want to get fancy, you can look into writing a "web service". But what I have outlined above is essentially that, albeit using an informal "protocol".