C# Collection for "most recently used"

I would suggest a LinkedList collection. The LinkedList holds references to the first and last node in the list (effectively letting you travel in LRU/MRU order). It also has AddFirst and Remove methods so you can remove the value and move it when a value from the list is used. MSDN

Efficient modelling of an MruList in C# or Java

Here's a Cache class that stores objects by the time they were accessed. More recent items bubble to the end of the list. The cache operates off an indexer property that takes an object key. You could easily replace the internal dictionary to a list and reference the list from the indexer.

BTW, you should rename the class to MRU as well :)

class Cache
    {
        Dictionary<object, object> cache = new Dictionary<object, object>();

        /// <summary>
        /// Keeps up with the most recently read items.
        /// Items at the end of the list were read last. 
        /// Items at the front of the list have been the most idle.
        /// Items at the front are removed if the cache capacity is reached.
        /// </summary>
        List<object> priority = new List<object>();
        public Type Type { get; set; }
        public Cache(Type type)
        {
            this.Type = type;

            //TODO: register this cache with the manager 

        }
        public object this[object key]
        { 
            get 
            {
                lock (this)
                {
                    if (!cache.ContainsKey(key)) return null;
                    //move the item to the end of the list                    
                    priority.Remove(key);
                    priority.Add(key);
                    return cache[key];
                }
            }
            set 
            {
                lock (this)
                {
                    if (Capacity > 0 && cache.Count == Capacity)
                    {
                        cache.Remove(priority[0]);
                        priority.RemoveAt(0);
                    }
                    cache[key] = value;
                    priority.Remove(key);
                    priority.Add(key);

                    if (priority.Count != cache.Count)
                        throw new Exception("Capacity mismatch.");
                }
            }
        }
        public int Count { get { return cache.Count; } }
        public int Capacity { get; set; }

        public void Clear()
        {
            lock (this)
            {
                priority.Clear();
                cache.Clear();
            }
        }
    }

Couple of comments about your approach

Why have Store return T? I know what I just added, returning it back to me is un-necessary unless you explicitly want method chaining
Refactor GetNext() into a new class. It represents a different set of functionality (storage vs. cursor traversal) and should be represented by a separate interface. It also has usability concerns as what happens when two different methods active on the same stack want to traverse the structure?
GetList() should likely return IEnumerable<T>. Returning List<T> either forces an explicit copy up front or returns a pointer to an underlying implementation. Neither is a great choice.

As for what is the best structure to back the interface. It seems like the best to implement is to have a data structure which is efficient at adding to one end, and removing from the other. A doubly linked list would suit this nicely.

You can build this up with a Collections.Generic.LinkedList<T>. When you push an item into a full list, delete the last one and insert the new one at the front. Most operations should be in O(1) which is better than a array-based implementation.