In short, Hime to Samurai is a good diversion for those wanting more of Trigger's SSSS. series. The spin-off takes place prior to the events of SSSS.Gridman and is completely unrelated to its story, serving mostly as a fun escapade and a way to showcase a little bit of Samurai Calibur, how he thinks and acts. The manga progresses through his interactions with Kudzuki Hime (the protagonist of the manga) and is very straightfoward, togheter they have to protect the city from rampaging Kaiju and find out why Samurai can't make contact with the other Neon Genesis High kids and why everyone seems to be forgettingabout the attacks as soon as the Kaiju are defeated.
Sounds familiar? The spin-off leans heavily on references to the main series, they know why you are here and they give it to you. From the memory erasing mist, to a self rebuilding city, to boxcutters, you'll certainly see lots of elements that will remind you of SSSS.Gridman here.

Art wise, the series is solid. While it has a slightly different style compared to the main series it still maintain the familiarity of every reocurring character. Action scenes and facial expressions are the highlights, with Hime being a delight to see with all her goofy faces. Kaiju and Barrier Gear designs aren't as clean as the ones you see in Gridman or Dynazenon in my opinion but are serviceable and don't detract in anyway.

I recommend any fan of the SSSS. Series to give it a chance, if anything it's quite short so it's not like it's a grand commitment anyway, but I honestly think people will find something to like here. If you don't mind spoilers, there is one more bit that I'd like to discuss, if you want to go in with as little info as possible stop here.

Spoilers ahead: If you're here, you probably really liked SSSS. Gridman and want more of it, and if you do I assume you enjoy the show for what it really is about and not just the "thicc thighs" meme. Well in this case I'll answer your main question, yes Akane is a part of the story, however don't go in expecting more of the broken antagonist of the series, but instead a completely different facet of her. I believe this Akane might not be to some's liking, personally I don't mind, it isn't better or worse, just a different take and I appreciate it for what it is.