There are two possible approaches.
If you can conceive a plan for moving Mount Fuji all in one piece — the way European monarchs had their
engineers move Egyptian obelisks to their capitals — good luck. Otherwise, it's a Fermi estimation. You treat moving Mount Fuji as if it were an ordinary excavation job. You need to estimate the volume of Mount Fuji in truckloads. The starting point of that estimation is likely to be the famous profile of Fuji. Most Americans have a mental picture of Fuji as a shallow cone,maybe five times as wide at the base as it is high. Most people have a much hazier idea of Fuji's height. Fuji is not in a class with the world's tallest mountains (Mount Everest being about 29,000 feet high). It is surely thousands of feet tall. Let's settle for the nicely round figure of 10,000 feet. (A lucky guess, for the summit of Mount Fuji is actually 12,387 feet above sea level.) That means we've got a cone 10,000 feet high and 50,000 feet across at the base.
Were Fuji a tuna can-shaped cylinder rather than a cone, its volume would be its base area times its height. The base is a circle 50,000 feet in diameter. A square 50,000 feet on a side would have an area of 50,000 x 50,000 feet. That's 2.5 billion square feet. But a circle has less area than a square of equal breadth (π/4 as much, or 79 percent), so let's say about 2 billion square feet. Multiply this by the 10,000-foot height, and you've got 20 trillion cubic feet as the volume of a cylinder exactly enclosing Mount Fuji.
Fuji is more like a cone, though. If you remember that a cone has a volume exactly one-third of the equivalent cylinder, give yourself extra credit. Whether you remember that or not, a cone obviously has less volume than the associated cylinder. Since we like round figures so much, let's cut the 20 trillion to 10 trillion and say that's the volume of Fuji's cone: 10 trillion cubic feet of volcanic rock. How many truckloads is that? A truck might be able to carry something like a 10-foot by 10-foot by 10-foot cube of excavated rock and soil. That's 1,000 cubic feet to the truckload. So Fuji represents 10 billion truckloads.
The question leaves a lot unspecified. We don't know where we're moving Mount Fuji. See if the interviewer will supply this information. We also don't know how much of the mountain is loose topsoil and how much is solid volcanic rock that might have to be excavated with dynamite. At best, the excavation and transport of a truckload is probably going to be a good day's work for somebody. Equating truckloads with worker-days, we estimate it will take on the order of 10 billion worker-days to move Fuji.
The project's time frame depends on how many people are on the job. In the implausible event that a lone individual is charged with the task (to be replaced successively, like lighthouse keepers, over the millennia), then it would take 10 billion days, which comes to something like 30 million years. (Fuji is probably not that old, and won't exist in its present form for that span of time. It would vanish through natural processes before one guy could move it.)
In the also-implausible event that all the world's 6 billion people pitched in (and had suitable equipment, and avoided getting in each other's way), you could move Fuji in a couple of days. Pretend that the Japanese government wants to move Fuji and marshals more or less realistic resources for the task. Ten thousand people, the size of a large corporation, might be a reasonable workforce. It would take them 10 trillion / 10,000 days. That's a million days, or about 3,000 years.