Know Before You Go
To visit the train, you'll need to purchase a ticket, which can be a small adventure in itself. The historical railway station where they were once sold has now been closed, but try the Beograd Centar station VIP or Information desks. Try to find a clerk that speaks English, and explain you want to see the "Blue Train." Failing this, the words in Serbian are "Plavi Voz."
It helps to email ahead using the address on the Serbian Railways' website under "Special Services." And, if all else fails, the folks at the Serbian Railway Museum in downtown Belgrade, some of whom speak excellent English, are very helpful in arranging a visit. Once you have a ticket, just hop on the #3 tram and ride all the way to the Rasadnik stop—the depot is just across the railway tracks.
Be cautious when crossing these and entering the depot--in addition to frequent train traffic the depot is actively used as a railway maintenance facility and there is frequent movement of heavy equipment. Show your ticket to the guard just past the main depot doors, and if they are open and there are some available workers, they'll open the train.