You can upgrade your Mini account to a shared hosting account at any time either through the client area or opening a ticket and request an upgrade.
When your upgrade is completed, the resources on your account will be upgraded, but not the resources for your specific domain. This is becasue some people want to upgrade to add more domains to the account. If you want to increase resources to the domain (disk space or bandwidth), you will need to log into the control panel > click on the domain you wish to manage > limits > and adjust the domain limits
If you don’t adjust your limits you will continue receiving email alerts.