with reference to that institution where this object is kept, when was that institution established?
1753
British Museum
The British Museum was established in 1753, largely based on the collections of the Irish physician and scientist Sir Hans Sloane. 
what is royal gold cup? It is a solid gold covered cup lavishly decorated with enamel and pearls. is it kept in a museum? Yes, in the British Museum who designed it? John, Duke of Berry was there someone from the royal family or anyone who utilized this piece? Charles VI of France will you find pictures of human figures in that object? Yes, symbols of the Four Evangelists run round the foot of the cup.
wiki:1541265
British Museum [SEP] Introduction
The British Museum, in the Bloomsbury area of London, United Kingdom, is a public institution dedicated to human history, art and culture. Its permanent collection of some eight million works is among the largest and most comprehensive in existence, having been widely sourced during the era of the British Empire. It documents the story of human culture from its beginnings to the present. It was the first public national museum in the world. The British Museum was established in 1753, largely based on the collections of the Irish physician and scientist Sir Hans Sloane. It first opened to the public in 1759, in Montagu House, on the site of the current building.