Java Singleton getter/setter
There are a couple bad things in the code

First of all, this is weird in many ways:

private Container() {
    System.out.println("Start");
}


You print out "Start" to the standard output. This basically restrict any usage of this class to have access to a console. What if it is used in a GUI application? What if this is used in a server application?

Also, "Start" what? Whoever read the console will have no clue what is started? 

The object is constructed as a static field, and Java did not guarantee when these static constructor are run (they only guarantee it is run before the class methods is run, so it is basically you have a console print that is shown in non-deterministic time.

This is not great too.

public void setAppName(String appName) {
    this.appName = appName;
}


Whoever got access to the Container object instance can call this method, which means you basically have a global variable that anyone can access. That's a bad idea, because it will makes debugging really difficult. 

From these classes, it appears to me that all you needed is to let the app name and app owner initialized once and get access anywhere, you can make that by simply having a Configuration class with a static property.

public class Configuration
{
    private static String appName;
    private static String appOwner;
    public static void Initialize(String appName, String appOwner)
    {
        Configuration.appName = appName;
        Configuration.appOwner = appOwner;
    }
    public static String AppName() { return Configuration.appName; }
    public static String AppOwner() { return Configuration.appOwner; }
}


That would be it, simple and clear.