The problem with answering this question isn't so much trying to figure out my first Anime/Manga focused campaign; that was a Mekton campaign from early high school, circa 1985 or 86. My first dedicated one without Mecha as the main draw was 'Blast City Blues', my high school Teenagers from Outer Space campaign from around 1987.
The only thing that doesn't add up here is that I was a big fan of the American broadcasts of Space Battleship Yamato as 'Star Blazers' and Science Ninja Team Gatchman or as we U.S. East Coasters knew it, 'Battle of The Planets', as early as 1979. I had seen other series dubbed in Spanish not long after, including Cyborg 009, Captain Harlock, Galaxy Express 999, and Captain Future. By the early 80s I knew that these shows were bastardized translations of animated series from Japan, largely thanks to my friend Aldrin 'Buzz' Aw who had seen them in his native land of Myanmar (at that time called Burma).
By 1984 and 85 I was sitting with club that would get together the second Saturday of each month to watch shows recorded off of Japanese television or on VHS release overseas. These show and films had no subtitles or foreign language dubbing. They were in Japanese for Japanese audiences and it was only thanks to synopses provided by the club's Japanese speakers that I had a solid idea of what was happening. Well, that and the excellent storytelling of those early series that made getting the gist of an episode much easier then I think would be the case today.
What I am getting at, what has me wondering is...what was my first game heavily influenced by my exposure to Anime and Manga. Was there a campaign that was strongly effected by my love of the medium, even if it wasn't really an 'Anime/Manga game'?
Answer that latter part first, yes, of course there was. Most likely it was The Winghorn Guard and its world setting of Aerth, my D&D-But-Not games. Aerth definitely has an Anime/Manga feel to it akin to what you would see in Record of the Lodoss War, Legend of Crystania, and other 'Western Style' Fantasy films and series from Japan, even though those would actually come out after the initial Winghorn Guard campaign.
I think the style and approach of these Japanese entertainment mediums had major influences on all the gaming I did, whether they were intended to or not, and that's why it's difficult to pinpoint a first time. My stories added more melodrama, romantic subplots, touches of comedy here and there, and intense and often over the top action sequences.
We're all the product of what we are exposed to and as I've stated before, my origins in the hobby are not extensions of Conan novels or re-reads of The Lord of The Rings. I'm far more influenced and inspired by Comic Books, Film, and Television and that most assuredly includes Japanese Comics and Animation.
It would be fascinating to take a time machine back to watch myself run games in the years between 1979 and the early to mid-80s to see if I could spot the point where Anime really started to rear its head.