No one I've seen has had a good idea of what is going on in this manga, so I would suggest that you read as much as you can stand of it and then start looking for other’s interpretations of whet they think is going on. (Note: other people’s input helps A LOT here) A few interpretation are presented latter in this review but they are very wordy and should only be attempted after you give up or finish the manga.

WARNING! GO READ THE MANGA SOME BEFORE PROGRESSING

For a story, Fushigi no Kuni no Shounen Arisu seems to hold a weak continuity and story line reusing the same characters to tell “love” stories between the Prince of Hearts and Arisu. Don't worry if you aren’t into man love ect. it isn’t that kind of manga, however you should be weirded out occasionally. This strangeness adds to the enjoyment and the overall ambiance of a drug induced hallucination that was the original Alice in Wonderland (Lewis Carroll was a Morphine addict). Hoverer, the plot is just an optical illusion, a mystery if you will, all the more dramatic when revealed at the end! The plot between chapters reads like a drug trip and is actual fascinating in that respect, understanding the action is underrated BUT if you just can’t stand it then find some one trying to interpret it and get an idea of whet other people think. This sort of ruins the mystery, like flipping to the back of a mystery novel so be warned. Each progressing chapter blurs the line between unrelated side stories, dream sequences and the “actual story” resulting in characters developing in story lines seemingly unrelated to the plot. All in all, the confusion seems to have subtle patterns indicating a rhyme and reason to the literature further confusing you. This strangeness is one of the reasons I think this is a great manga. Anything that gets you going “What the heck was that!” hours or even day later is a true work of art, maybe it won’t be enjoyable but it will always be fascinating and I think this makes Fushigi no Kuni no Shounen Arisu a fantastic read.
The other reason I so greatly enjoy the series is that I think Arisu becomes a great tortured character. Arisu’s overall personality and the sort of “self hate that should be developing but isn’t” as he progresses through the story line are my favorite way that writers mess with characters. His agony and confusion in latter stories as everything becomes subtly wrong to him, even himself and eventually when he just breaks down because everything he ever knew or believed becomes wrong and the only thing left in his faded memories stretching back as far as he can remember is the Prince of Hearts. That is some sad stuff, but great reading. Stockholm syndrome for the win!

What is actual strings the chapters together, for everyone who just doesn't get it. Its ok if you have read the manga three times and asked your friends and forums and you still don’t get it. This is the natural order of things, sit down with a dictionary and I will explain.
Pre work: do not begin reading until you understand the following terms!
Dimension/ Plane of Existence
“Alternate” or “Parallel” Reality
Lucid dreaming

#Idea 1 No continuity: nothing in any of the chapters can be thought of as effecting any of the other chapters, if it seems like it does then it is just to mess you up.
#Idea 2 More than one Arisu: Every time Arisu enters Wonderland or returns home, he never does, in fact he shifts to one that is almost, but not quite, the same as the last one he was in which was just vacated by another Arisu. This is an adequate explanation and what gave me my final aha moment, answer #3.
#3 Power of a God: Arisu doesn't know it, however due to the nature of wonderland as explained by Alice and shown in chapter two he has the ability to change wonderland with only his thoughts. When he retuned, he somehow took a bit of wonderland with him and though the manga as it is demonstrates that he can return to Wonderland, he subconsciously forms the wonderland and bases what he is creating on his own world view and recreates himself a bit with it. This normally wouldn’t be too bad, he would end up taller, handsomer and a better singer than he formerly was, but no big deal. However, the constant bombardment of affection from the Prince of Hearts causes a problem, all of those little things that you do that are out of character and dirty little thoughts that you have combined with the Prince’s attempts at Arisu and the general chaos are stacking up into a knock back combo pushing Arisu from what he was to what people (the prince) think he is. It certainly doesn't help that each jump smears Arisu’s memories eventually leaving his only constant input as “You are Alice, bride to me the prince of hearts! Come away with me and we will be married”.
Following this mental recreation of everything creates the following changes and stationary concepts over the chapters. The Prince of Hearts gets more manly going from crybaby to playboy to finally loving man as Arisu tuns him into the manly man he can love. Arisu slowly shifts from “the Arisu who got on the bus” to “the Alice that is to be wed to The Prince of Hearts” because this concept has become a solid rock in an ocean of confusion and change. Also fairly constant are “Arisu is a man, he live here” and “The White Rabbit”. What is really going on in the end is that Arisu dreaming, fantasizing, what ifs, self doubts, and the Prince’s superior force of will are destroying Arisu’s stable image of reality causing Arisu to create reality as he thinks it is, unstable thus creating more chaos. Repeat this cycle 2 or 3 times and you have the manga.
You may also want to decode individual chapters and for that you really need to put them into any of three categories: Reality, Lucid Dreams and Unconscious Dreams. Reality is what you think of as the core of the continuity he is not in wonderland and he is Arisu. Lucid dreams are chapters were Arisu has total memory and is in wonderland or an actual dream, everything make comparative sense but has an unrealistic tinge. He remembers these experiences fairly well and they are always as much of an influence as reality is either directly or though reshaping of Arisu’s world view. Unconscious dreams are those side stories that seem to not be part of the continuity at all. These are where Arisu creates a world that isn’t related to the story, plays out a story and then tweaks the characters based on what you see in that story. If these are real honest to god dreams or trips to wonderland with everything made weird, I don’t know, but everyone has a new history for the course of the dream and never remembers it latter on so even if it is just as real as everything else the only part of it that matters outside of the dream is that little part that is subtly altered because of the dream.

I hope this gave you some new eyes to try and reread the manga with or if you were bad and read all of the way through convinced you to read it once. It is worth the examination.