I certainly would not want to trade places with anyone from that time period because I would not want to leave behind the ease of modern life. I would not want to be sent back to a time before technology made our lives so easy.
However, there would certainly be benefits to living in the Mongol Empire, so long as you were not killed when you were being made part of that empire. It is said that the Mongols maintained a society that was safe and stable. They prevented crime and lawlessness within their borders. In addition, they allowed people of all religions to keep their faiths. This continued so long as the people were obedient, but punishments for rebellion and other crimes were harsh.
So I would not want to have to live back then, but the Mongol Empire was at least a relatively safe and stable place to live if you had to live back then.