Is this an actual production website? Is it important that it's up most of the time? If so, then no, it's not possible for you to manage a production webserver with no linux experience.
At a minimum, you need the following:
- Dependency software installed (MySQL/Apache/etc)
- Application software installed (Wordpress, whatever else you are running)
- Monitoring (you need to know how loaded your box is, what kind of requests/sec you are seeing)
- Regular maintenance (Keeping up to date on security fixes, installing new software)
- Backups (no, it's not a mistake that this is listed three times)
- Knowledge of what to do if/when a security breach occurs
Simply put, a production web site is not the place for you to be learning your linux skills. I'd suggest you get a VPS and run some personal or development websites on it, in order for you to start learning about it. Don't attempt to go from a managed hosting environment to unmanaged unless you know exactly what to expect. Based on the type of question you asked, I wouldn't suggest you move to an unmanaged VPS yet.