You only need the lock if you are doing more than one operation against the application state. In you case you are just doing one operation, so it's perfectly safe without the lock:
Application.Set("Name", "Value");
If you do more than one operation, and they rely on each other, you need the lock. For example:
Application.Lock();
string name = Application.Get("Name");
if (name == null) {
Application.Set("Name", "Value");
}
Application.UnLock();