Protection and sexual selection
Before we begin with what'd be necessary for a humanoid to live comfortably in a desert environment without our "cheats" , let's first look at what it takes to live there:
To survive in a Sandy desert environment, the main adaptations needed are to be able to function in both extreme heat and extreme cold (Sandy deserts can easily reach negative temperatures at night) and to loose as little water as possible. When it comes to humans, we make wide use of cotton clothes that cover a lot of skin, offering protection from the sun and the sand and helping to retain sweat in order to help them cooling down. By nightfall, the same cotton clothes help retain heat.
As for creatures that are actually capable of surviving without the use of clothing, let's take a look at the desert specialists: camels.
Camels are tough cookies when it comes to surviving in deserts: they can easily tolerate both extreme heat and cold, they're adapted to the extreme to be able to both function on very little water and to loose as little of it as possible, they have furry skin with leathery patches for resting in the sand without getting burned. Their eyelids are thin and mostly transparent, so they can see during sandstorms. Their feet are also wide in order to walk through the sand without sinking.
So basically, our mummy ideally needs at least all of the following:
the ability to keep themselves cool during the day and warm during the night.
an extremely water-efficient body.
at least some patched of tough leathery skin in order to walk and lie in the sand without getting burned.
wide feet for walking in the sand.
And for our mummy we need the following:
My best bet in such a situation would be to go down the armadillo game: a heat that's covered by colorful armored plates, formed through sexual selection that cover the face and head, with horn-like growths on the sides forming the characteristic crest. If could resemble a mask as long as they keep the mouth shut, otherwise it'd probably be an unsettling sight.
As for the blue markings, if could be potentially explained through the presence of microscopic structures in these armored patches in order to create the illusion of a blue color, while the rest of the plates is pigmented in yellow. This armor and the coloring could be explained by both defensive and sexual selection, as vibrant blue colors in nature can often signify you're poisonous, and the special patterns may also help attract a mate of the same species. They'd probably still have noticeable thick eyelashes, and the transparent eyelids could give the impression that they sleep with eyes open to someone less attentive.
The bandages part is trickier. The first thing that comes to mind would be a leathery, wrinkly skin, which resembles someone covered by bandages well enough and covers both the burning problem and the heat dispersion one somewhat. This is useful for loosing heat, as it amplifies the surface area, but this same mechanism is terrible for a cold environment where you don't want to loose heat. The reason why making a loose clothing out of your own skin here is precisely that, being still part of your body, it would drastically increase your surface area compared to your volume, making cold environments potentially deadly to say in for too long.
A second, likely more functional alternative would be for them to be, like camels, mostly covered by thick fur in order to offer both protection from the sun and heat as well as insulation during the night, while having a skin that's fairly leathery in order to protect then from burns. The fur could be patterned as to resemble someone wrapped in bandages from afar (perhaps as a result of sexual selection), but the illusion would be easily broken once you get close enough. At best, they could have patches of loose skin resembling the ends of bandages in some spots of their body such as near the hands, helping somewhat to loose heat during the day by slightly increasing surface area and getting pulled closer to the body through muscle contraction at night, helping with the illusion that the person is wrapped in bandages, some of which are coming loose and hanging near the extremities.
The remaining changes are mostly related to internal structures and thus not as noticeable. Perhaps they're more like chimps and have an omnivorous diets, with larger mouths, stronger Jaws and a wider gape. Having physical strength akin to a chimpanzee's, although with a more bipedal structure, could also be helpful for capturing prey.
Overall, I'd say it's not an easy task, and a more realistic mummy sapiens would be strangely soft to the touch and with wider feet. Like camels, they'd be able to survive both in very warm and very cold places, and an omnivorous diet would help ensure they can get the food they need more easily. A less developed brain would probably also help here, as a less costly brain would mean their fat reserves can last longer. They'd probably travel long distances in groups, searching for food and water. They would however be at a great disadvantage when it comes to hunting, as the colors of a mummy with a funerary mask aren't the most conspicuous, and don't exactly blend in with the colors of the desert.
