Okay, so that's what I was afraid of, and it's why I wrote what I wrote...We can help you through this, but you're going to have to likely buy some things or try to find some things, but in Dubai it shouldn't be an issue, you should have access to buying everything you need.
No, you cannot keep him in a box with a light on him!!! At only 1 month old he does not yet have his outer-feathers grown-in, so he MUST be kept in a constant ambient/air temperature between 80-85 degrees F until ALL of his outer-feathers grow completely in. So you must make a homemade Brooder.
(putting a light over a bird in a box not only will not keep the air temperature warm enough, but having a light on them all the time is bad for their eyes, and depending on what type of "light" it can hurt their skin and feathers as well...And in-order to get the temperature inside the box warm enough, you'd have to have the light really very close to your bird, and that can cause blindness, skin damage/burns, and a bunch of other issues...Parrots aren't like poultry-birds, like baby chickens who you can put under a heat-lamp 24 hours a day, and even chickens need a special type of heat-bulb made for birds...
***What you need to do is find a cardboard box or a plastic box that is large enough so that it can have a "Warm-End" in the back-half of the box, and then a "Cool-End" in the front of the box. Then you need to buy, if you don't already have one, an electric heating-pad that you can adjust the temperature of (usually they have low, medium, high settings, etc.), just an ordinary electric heating-pad that they sell for people. Then you place the BACK-HALF of the box right on top of the electric heating-pad, leaving the front-half of the box off of the heating-pad. Then you need to also buy, if you don't already have one, a digital or analog, doesn't matter, ambient thermometer that you will place in the back-half of the box (just set it in the bottom of the back-half of the box). You can buy a digital thermometer at most Pet Stores, they sell them for Reptiles and for Fish in pretty much every Pet Shop, or you can use one that is meant for people to use to tell the temperature outside, as long as it is small enough to be set in the bottom of the back-half of the box...This is what needs to always say between 80-85 degrees F so you know that your bird is always at a warm enough temperature so he doesn't become extremely sick with a Fungal Infection in his Gastrointestinal Tract...Then you also need to keep a good inch or two of some kind of "bedding" in the bottom of the entire box at all times, something like the type of bedding you buy at the Pet Store for hamster, guinea pigs, rats, etc. The kind that is made out of shredded-paper is the best, either that or some kind of bird-safe wood-chips (NO CEDAR CHIPS, they are too fragrant)..Just don't use anything like Corn-Cob or Walnut-Shell bedding, as your bird can choke on them. The reason you have to have the entire box lined with bedding is because at your bird's age he can still develop "Splayed-Legs" if you keep him on just a flat surface, like the bottom of the box, because the bedding keeps their legs pushed together when they are standing/sitting and walking. And it also will absorb his droppings of course, and you need to change out and replace any soiled bedding once a day....Then finally you need to cover ONLY THE BACK-HALF OF THE BOX THAT IS OVER THE HEATING PAD AND HAS THE THERMOMETER IN IT WITH A TOWEL OR BLANKET,, because this will help the temperature in the back of the box stay within the 80-85 degrees F at all times, while leaving the front-half of the box uncovered.
Then you just turn-on the heating-pad to low, cover the back-half of the box with the Towel/Blanket, and wait about 15-20 minutes and then check the temperature on the thermometer in the back-half of the box...If it's not yet reached between 80-85 degrees then you need to bump the temperature of the heating-pad up another setting and then wait another 15-20 minutes and check it again. Usually it takes the "Medium" or middle-setting on the heating-pad to get the temperature up to between 80-85 degrees F. And that's a homemade Brooder with a warm back-half, where he'll spend most of his time for the next few weeks, and a cool front-half that he can go to if he wants to. And that's the other reason you cannot just put him "under a light", not only will he not be warm enough and will the light only heat one particular spot up, which is dangerous for his eyes, skin, etc., but it also doesn't give him the option of a cooler-area to go to when he needs to. MAKING A BROODER THAT IS KEEPING HIM AT 80-85 DEGREES F IS NOT OPTIONAL, HE WILL BECOME VERY SICK WITHOUT THIS OR IF YOU JUST PUT SOME KIND OF LIGHT OVER HIM...And then what happens all the time is people come here when their birds are already sick a couple of weeks later, and they have to find an Avian Vet for the correct medications to save the bird's life...
****What are you feeding him for formula? And are you using a cooking/candy thermometer with a probe you can keep in the formula at all times to make sure that any formula you are feeding him is between 104 degrees F and 110 degrees F? If not then this is going to make him very, very sick, especially if he's not being kept at the correct ambient temperature either! You must have a digital candy/cooking thermometer with a metal-probe on it that you keep in the formula at all times while you're feeding your bird!!! Usually people in your situation just use warm/hot water from the sink to mix their formula with, and it's nowhere near hot enough to get the formula up between 104-110 degrees F, which always results in again a Fungal Infection...Or they just heat water up in the microwave or on the stove and mix the formula up and "test it on their wrist", which you just cannot do!!! You will either make him sick with a Fungal infection if it's too cool, or you will burn his crop if it's too hot, and then again you'll have to find an Avian Vet immediately once he gets sick, or once you see his Crop develop a wound on it/in it...
At least you've only had your baby Alex for a few days, you still have time to correct his ambient temperature and his formula temperature, and hopefully he'll be fine...But you need to get these things taken care of immediately, because this happens at least a couple of times a month here in the forum with unweaned, baby Alexandrines who become sick because they are not kept in a Brooder/homemade Brooder, they either just have a lamp over them or nothing at all, and because they are not measuring the temperature of the hand-feeding formula...Once a Fungal Infection starts, his Crop will stop emptying properly, which causes him to start to refuse the formula. This is called "Slow-Crop" when it's not emptying as quickly as it should (in the early stages of a Fungal Infection), and within a week or two it turns into "Crop Stasis", which means the Crop just stops emptying all together, and by that time it's often too late to help the baby bird...So it's very, very important that you get your baby Alex into a Brooder-box as described, and that you get a cooking thermometer and you keep it in the formula at all times while you're feeding him, and if the formula temperature drops below 104 degrees F while you're feeding him, you have to go and heat-up some more water, mix it into the formula very well, and then once it's under 110 degrees F again but above 104 degrees F again, then you can finish feeding him...
***The other issue that we often see here with these unweaned Alexandrines is that they aren't being fed a baby bird hand-feeding formula, like Kaytee Exact, Roudybush, etc., but rather a human-baby formula or cereal, such as Cerelac or Similac...If you are not feeding your Alex a hand-feeding formula made for baby birds, but rather a HUMAN BABY formula/cereal (usually it's Cerelac that is used), then you must stop immediately and get some proper hand-feeding formula, because Human Baby Cereal/Formula like Ceralac is LOADED WITH ADDED IRON, and birds cannot handle eating much Iron at all. They develop a serious medical condition called "Iron Storage Syndrome/Disease", which is fatal if it's not caught early-on and the Iron is stopped. Human Baby's need tons of Iron, so they purposely put a ton of Iron in Cerelac and all other formulas/cereals meant for human babies, but birds just cannot process it, they can't store it like we do, and it causes them to become very seriously ill when it's given to them in huge amounts like are in Cerelac and the like...Once it builds-up in their bodies, they first start vomiting, then they start showing neurological signs/symptoms such as Tremors, then Seizures, and finally they start having trouble breathing and they die. So hopefully you're using a proper baby bird hand-feeding formula...The other issue is that human formula/cereal does not contain the proper amounts of ANY of the vitamins, minerals, amino acids, proteins, fat, carbs, etc. that baby birds need to develop properly, and they often don't grow correctly, are stunted, and have all kinds of health issues throughout their entire lives if they are fed nothing but Cerelac or other baby cereals/formulas as babies. This is the second largest problem we see with baby Alexandrines who are sold to people unweaned and too young...
I know it's a common practice in other countries to buy baby parrots as very young and unweaned, but the problem is that without the proper knowledge, supplies/equipment, and the proper diet, they die. And we see baby Alexandrines becoming sick and dying every month here in the exact situation you're in...So hopefully we've caught these issues early enough that your little guy will be okay once they are corrected.