I would say, based on what a Shinigami does, it would be a warrior/magic-user combo that specializes in defensive combat versus spirits and ghosts.
After all, A Shinigami escorts spirits to the after-life. There are likely plenty of nasty spirits (Or similar) that would love to get their incorporeal hands on a lone spirit.. so a Shinigami must be able to protect them. Since the Shinigami don't want to go out and destroy spirits, most of their abilities would be defensive against spirits..
Of course, since Shinigami are psychopomps, they probably have a quota of souls that they must deliver to the afterlife. And a higher-up to report to.
Following tradition, various genres of fantasy would be rather distant from each other, so a Shinigami would be something of a foreigner in the generic medieval fantasy land, from generic medieval oriental fantasy land.
Which would be both advantageous and disadvantageous. On one hand, none of your foes would know what exactly you were, and thus find it hard to know your strengths or weaknesses... on the other hand, no potential allies know what you are either, leaving them free to misinterpret you as something bad or hostile.
“She'd become a governess. It was one of the few jobs a known lady could do. And she'd taken to it well. She'd sworn that if she did indeed ever find herself dancing on rooftops with chimney sweeps she'd beat herself to death with her own umbrella.”
― Hogfather