While there were several kinds of art collections during the middle ages, the earliest museum (having its own dedicated building) was the museum of Ambras Castle, started 1570. Here the term "museum" was also being used for the collection.

If that's too far from the middle age period for you, you could research the art collections of the capitoline museums (started in 1471) or the musei vaticani (started 1506). But keep in mind that these did not have their own building, which doesn't make them count as proper museums in my opinion.

Museum Ambras started in the 1570s with Ferdinand II collecting a variety of armour. The collection was not available to the public per se but there are records of guided tours in return for payment.

The museum still exists today by the way, so it's not only a museum of armour (and much more) but also a museum of the first museum. :)

Ancient powers would collect artifacts from groups they conquered (famously the Romans collected spolia as in the dozens of obelisks taken from Egypt). In some instances these “spoils” were placed together on display and have interpreted this as early “museums”. The earliest example of which I am aware is at Susa where the Stela of Naramsin and Code of Hammurabi (all now in the Louvre) were found.

Edit: for more on this inscription, and other relevant discussions, read this great article FELDMAN, MARIAN H. “Knowledge as Cultural Biography: Lives of Mesopotamian Monuments.” Studies in the History of Art 74 (2009): 40–55. http://www.jstor.org/stable/42622715.