It's been established in another post the weights of Pegasi, unicorns, and Earth ponies, but one there's one thing I haven't figured out that's related to that: how much do alicorns weigh?
Unlike the average pony, Alicorns vary greatly in size
While we can somewhat estimate the size of the average pony (because so many use the same base model), every Alicorn we've seen in the show is noticeably different in size. So we can't really define how much "they" weigh as a subspecies.
Furthermore, while there have been attempts to nail down even standard pony sizes, they are often based on physics and scales that aren't always consistent. To come up with an accurate weight range for Alicorns, for which there is greater variability, less sample size, and hybrid physiology, is highly unlikely.
But if you want a ballpark estimate, the lower bound is "slightly more than the average pony" (Twilight's size). The upper bound is somewhere around Big Mac, since while shorter than Celestia, he's also more bulky.