Was it true that Emperor Wudi from the Western Han Dynasty attacked the Mongols and failed, causing the fall of the Western Han Dynasty?

It is not at all true that Han Wudi attacked the Mongols and failed. Han Wudi was successful in attacking his neighbors and extending Chinese control in every direction. He ushered in the golden age of China's history.

If you consider a timeline of events for all of the players in question it becomes obvious that this question is false. The Mongol Empire existed between 1206 and 1368 AD. Han Wudi was the emperor of Han China from 140 BC-87 BC. The Han dynasty of China survived the death of Han Wudi for another three centuries. When you look at this chronology of events, you realize that Han Wudi died thirteen centuries before Genghis Khan and the Mongol Empire. Also, keep in mind that the Mongol Empire was the aggressor against China, not the other way around.