It would die. Pretty quickly, in all likelihood.
If you’ve never heard of the square-cube relationship, allow me to introduce you. If you scale up an object by, say, a factor of 100, the volume of the object increases by 100³, or 1,000,000, but the cross-sectional area increases by 100², or 10,000. In other words, the volume increases 100 times more than the cross-sectional area. That relationship holds for whatever scale factor you choose. So if you scale an object up by a factor of 200, its volume increases 200 times more than its cross-sectional area.
Why is that important? Because the weight of a critter is directly related to its volume, but its strength is directly related to the cross-sectional area of its muscles, or whatever muscle analogue it has.
The typical adult household cockroach is about 4 cm long, and the average adult human is about 165 cm tall. In order for a cockroach to become as long as a human is tall, it would have to scale up by a factor of 165/4 = 41.25. Assuming the same density, the mega-roach would be about 70,000 times heavier than the original, but it would only be about 1700 times as strong. In other words, its strength:weight ratio would have decreased by a factor of 41.25.
If a normal adult cockroach is able to lift 41 times its own body weight, then the mega-roach might just be able to get up and move around. However, I doubt that would be the case. I went looking to find out how much weight a normal roach can carry. The only reference I could find said that a cockroach can lift about 20 times its own weight; however, that information comes from another Quora answer. The person who gave the answer did provide a link, but the link seems to be broken.
Insects: How weight much can a cockroach lift?
If that person’s answer is accurate, then our human-sized roach won’t be able to move, let alone run. It won’t be able to get food, but starvation is not going to be its cause of death.
Cockroaches don’t have the same kind of respiratory system that humans and other large land animals have. We humans breathe oxygen into our lungs, where it diffuses through moist membranes into the bloodstream. From there the oxygen is transported to the rest of the body.
Cockroaches do not have lungs. They inhale air through body openings called spiracles. The spiracles open into a network of tubes called tracheae, which deliver oxygen-rich air directly to the cells. The oxygen diffuses through the walls of the tracheae into the cells. This works fine when the cockroach is small, because every cell is within a few millimeters of a tracheal tube, and oxygen can easily diffuse the short distance from the nearest tube to the cells that need it.
If a cockroach were to dramatically increase in size, many of its cells would be too far away from the nearest tracheal tube for oxygen to reach them. Without oxygen, these cells would start to die off. The insect would die of hypoxia long before it died of starvation.
Even if a cockroach could survive as a human-sized monstrosity, it wouldn’t eat us. Cockroaches are scavengers — they only eat dead and decaying organic matter. They don’t kill and eat living organisms. A human-sized cockroach wouldn’t become the scourge of humanity — it would feed on our scraps, just like it does now but at a much faster pace.