To speak about this OVA it's important to acknowledge first of all the huge difference between the manga of Kamisama Hajimemashita and it's anime adaptation. The adaptation has never been completely faithful to the manga, and the same happens with this OVA. If you haven't read the manga, this OVA is just a great addition to the anime. However, if you already read the manga, you won't be able to help feeling a little disappointed.
The OVA is very well done, just like the regular anime; good art, good cast, good rythm, good everything. The love between the main characters is reinforced and some secrets are revealed. BUT! (And this is a huge "But") The feels one gets in the manga are 300% more intense than in the OVA, so as an adaptation, it's just mediocre.
I'll try to explain briefly what we are missing in the OVA without giving any explicit spoiler:
First of all, the time-travel logic is far better depicted in the manga, and Nanami goes through way more hardship to accomplish her mission. Besides, with regard to the whole story, it is in this Past Arc where we are supposed to discover most of the important information about the nature of Tomoe and Nanami's relationship, as well as Tomoe and Yukiji's, which is not explaind in the OVA at all, when it really shoud have been because it is by far the most interesting part of the story and it really changes everything. (It is there where you finally say "Hey! This is clearly not Inuyasha!" ) But not trace of it in the OVA.
Last but not least, within this arc of the manga, Nanami gets to know quite well Akura-ou, who becomes a major character in the following arcs, and the absence of the Akura-ou part in the OVA, added to the fact of condensing everyone's favorite part of the manga (4 volumes of it actually) in barely 4 episodes, only makes me fear that there might not be next anime season in the future.
The past arc of Kamisama Hajimemashita has been one of the most pasionate stories I've read in a manga, and one of the most coherent time-travel stories I've ever come across. So it saddens me considerably to see it cheapened like this.
However, as I said at the begining of the review, if you haven't read the manga, go ahead, the story is undoubtably enjoyable and sweet.