Sugarcane juice would NOT be a "substitute" as it's what the original recipe for the Master Cleanse called for. Stanley actually had to find a substitute for it since it was not readily available in the US.
Probably any kind of pepper you can find in the capsicum genus would be fine. Whatever you do find you shoud be able to find it's genus with a little research on the net.
One place to possible start to search is using one of the translating engines on the net and try translating cayenne into Kuwaits native language which I assume is Arabic.
Paprika is pretty mild heat wise and it's made from dried red bell pepper which is a rather sweet and mellow fruit. The heat factor is definitely part of the the desireable theraputic property of the cayenne as it stimulates circulation and losens mucous in the system. I would think that because of its mellow nature paprika would not be suitable.
Cayenne is more of a red chili pepper. I tried I-Google translation engine English to Arabic for caynne and it returns the translation in Arabic text . Try that and see if folks in Kuwait can point you to it's equivalent. Try translating red chili intead of cayenne. Surely the Arabs must have some close equivalent. It could simply be just a matter of what they call it there.
On the cane juice I assume that you know that the stuff does not keep overnight even if refrigerated so you would have to have made fresh every day.
Another way to tell that the can juice is going bad is also color. It will start geting a slight brown tinge. Quite possibly if you mix the lemon juice into your batch it might act as a preservative and help it last a little longer. I can't say the preceding is "fact" since I'm just going from vague childhood memories which I seem to remember my mother adding lime juice to it to make it keep a little longer.
I've recently moved much closer to the farmers market where I found fresh cane juice in the states (Atlanta, Ga). On a recent visit last week I checked with the prospect of doing an MC with it in the spring but unfortunately they don't have it any more. Dunno that I could have afforded to do a full MC since it was fairly expensive at $6-7 US for half gallon but at least it would have been a nice occasional treat and change of pace kinda thing.
Though my experience is sparse in the last 20 years I don't remember seeing "fresh juice" with brown coloring and only seeing it get the brown color as it oxydates/ferments. Could be the extraction method same as veggie juicers where the ones that preserve the veg juice longer work at slow speeds and the faster spinning ones both generate heat and introduce oxygen into the juice making it spoil faster.
Ya that's sorta the kind of machine I remember from my childhood. They were great big things too with their own stands etc. They were enclosed in stainless steel and they had a softball sized hole in front that you could see the rollers but not the rest of the mechanical works. The cane came out the rear into a waste bin or trash can. They only ran the cane through it once and the outside skin was was peeled off pre juicing which to my my thinking makes it more sanitary. This running it through only once and peeling could be the difference.
In more recent years the ones I've seen are table top models. Not quite as small as a veg juicer but not anywhere near those old behemoths I remember from my childhood. With these machines the cane was also peeled and run through only once.
Ok I imagine that this is more like what you describe from India. For me it's a totally different experience. The machines I'm familiar with were inside restaurants/coffee shops in Cuba and in Miami where the health regulations I'm sure were way more strict and the machines were enclosed in stainless steel covers and all you could see were the feed rollers through a softball sized hole in the stainless enclosure. The internal works were a bit more sophisticated where it used 3 rollers which bent the cane as it crushed it making it more efficient thus not having to put the cane through more than once like the embedded video below from India. There are other vids on You Tube that show the same kinda machine but hand cranked.
Well I coulda picked amongst many videos on YouTube and different machines. I just happened to pick the one I posted more for the gist and the mechanics of your description of the "old fashioned washing machine rollers" you mentioned. I smiled at your video cause that guy was sure gonna get every damn last bit of juice from that cane as may times as he folded it and put it through again and again ;+D
Here is a western encased cane juice machine with easily removable rollers for washing to meet the more stringent western health code requirements. This is a table top model as you can tell from the dimensions. The ones I remember as a kid were like 3ft stainless steel cubes that stood on their own legs. The one in your vid looks like it could be at least 50 or more years old and quite possibly at one time was hand cranked and they just added a pulley with a motor.
He might not drool so much when he found out this little table top model was I believe around $1700 US