How can I get a message bundle string from inside a managed bean?

You can get the full qualified bundle name of <message-bundle> by Application#getMessageBundle(). You can get the current locale by UIViewRoot#getLocale(). You can get a ResourceBundle out of a full qualified bundle name and the locale by ResourceBundle#getBundle().

So, summarized:

FacesContext facesContext = FacesContext.getCurrentInstance();
String messageBundleName = facesContext.getApplication().getMessageBundle();
Locale locale = facesContext.getViewRoot().getLocale();
ResourceBundle bundle = ResourceBundle.getBundle(messageBundleName, locale);
// ...

Update: as per the mistake in the question, you actually want to get the bundle which is identified by the <base-name> of <resource-bundle>. This is unfortunately not directly available by a standard JSF API. You've either to hardcode the same base name in the code and substitute the messageBundleName in the above example with it, or to inject it as a managed property on <var> in a request scoped bean:

@ManagedProperty("#{msg}")
private ResourceBundle bundle; // +setter

FacesContext context = FacesContext.getCurrentInstance();
ResourceBundle bundle = context.getApplication().getResourceBundle(context, "msg");
String message = bundle.getString("key");

here is key is property name which you want to access from properties file .

       message = This is "message"

This entry is from messages.properites file. and "message" is "key" .