The Uganda railways has not operated for transporting people since the early 1990s. The only exception has been a train that connected Nairobi and Kampala on a once-off trip for an international qualifier for the World Cup Soccer in 2010.
The Kampala train station, though, has been kept in shape and, if you show up at the right time and talk to the right people, you can even get a tour of building. You probably will see the train that was used to transport soccer fans to and from Nairobi.
Besides bad management, the primary reason trains have not been running is that the international bus companies are in the hands of MPs, who personally financially benefit from the train network not being available.
All that said, there's a tiny bit of hope. The Chinese have showed an interest in financing some rebuilding activities, the Kenyans are very interested to connect their ports with Uganda, South Sudan, Rwanda, and even the DRC, and last year, for the first time in over a decade, a goods train connected Kenya with the far north east of Uganda, carrying material to be shipped to South Sudan. This, after that particular rail line had been refurbished during the year before.
Much of the tracks that lead west from Kampala, though, have been damaged or disappeared beyond repair.
So, to answer your question: No, it is currently not possible to catch a train in Uganda. But, give it a decade and this might have changed.