You need to be more precise about "the object". If you modify the object instance's properties, simply storing it again will perform an update:
MyClass someInstance = ObjectContainer.Query<MyClass>().FirstOrDefault();
someInstance.Name = "NewName";
someInstance.PhoneNumber = 12132434;
ObjectContainer.Store(someInstance); // This is the update call
[This is just pseudo-code]
So you don't need to match objects to each other as you would have to when using an RDBMS.
However, you need to make sure you are not using a different instance of ObjectContainer, because a different container will not know these objects are the same instance (since there is no ID field in them).
Your application architecture should help to do this for most workflows, so there should be really only one IObjectContainer
around. Only if timespans are really long (e.g. you need to store a reference to the object in a different database and process it somehow) it'd use the UUID. AS you already pointed out, that requires to store the ID somewhere else and hence complexifies your architecture.
If you however intend to create a new object and 'overwrite' the old object, things get somewhat more complicated because of other objects that might refer to it. However, this is a somehwat pathological case and should typically be handled within the domain model itself, e.g. by copying object data from one object to another.