I think System Admin experience will be considered for testing (not sure about this). If you have been working as a System Admin, you have good configuring and troubleshooting skills. With troubleshooting skills comes the skill of looking at the right place for the right thing which I think will be helpful software testing. Why don't you try working as a System Admin, Linux Admin or Database admin? I think your experience will be more relevant to those roles.
Programming is not mandatory for testing but having programming knowledge will be an advantage. To know about the top software testing tools, visit this link.