It's a bit of a drag at first, being an episodic show. Each episode deals with Fumika, our protagonist, delivering the titular shigofumi to someone. The episodes will typically deal with the circumstances around that; shigofumi are, after all, letters from the dead. However, with each episode, there are slight plot teases which work to build up the idea that something is going to happen.
The mystery is introduced fairly on - and mentioned in the AniList synopsis: Fumika is the only Shigofumi postman who ages. I think this is first brought up in episode 2. Guesses can be made at why around episode 5 or so, and it's outright stated in episode 8 or so. However, the mystery, I'd say, is only one small part of the show and its appeal. Once the mystery is solved, the viewer's investment shifts immediately to all the circumstances around it.
The series really kicks it into overdrive in the second half, with all the plot teases finally coming together to form something strong and coherent. I went from taking the series somewhat casually to actually deeply enjoying it in the second half. The plot progression and the "fallout," so to speak, of said progression was a lot of fun to follow and really elevated the series from that point forward.
One thing that I really appreciated was that the series turned out to be something different than I expected. While it wasn't a complete tonal shift like something such as [meta]
Madoka or SukaSuka,it definitely went in a direction that appealed to my personal tastes. The feeling was more akin to not caring all that much about something because of the perception it gives off, and then being pleasantly surprised when it catches you off guard.
Shigofumi's characters are fairly solid. While they're not strong enough to be the driving force behind the show, they certainly aren't weak enough to drag it down, either. Fumika has a certain attitude that makes one go "ugh, of course" at first, which shifts to an "oh, I see" as the story plays out. Her talking staff, Kanaka, and the running gag of her desire to be human is a humourous, pleasant distraction from some of the more somber tones and themes of Shigofumi.
Conclusively, I don't think that Shigofumi really does anything outstanding. There's nothing awe inspiring or breathtaking about the series. Conversely, however, it achieves what it sets out to do, and it does it well. Rather than the feeling of having stumbled upon a new favourite, Shigofumi leaves the viewer with a feeling of satisfaction. It invokes that happy sigh that comes after finishing a great novel; there's no need to spread it around, nothing in your life has changed, but at the same time, you're still glad you did it.