This is a fairly simple solution:

init' :: [a] -> [a]
init' ls = foldr (\x xs n -> if n == 0 then [] else x : xs (n - 1)) (const []) ls (length ls - 1)

Of course, init' will fail on infinite lists. Tail can be defined using foldr too, though:

tail' :: [a] -> [a]
tail' ls = foldr (x xs (y:ys) -> ys) id ls ls

takeWhile and dropWhile can be defined similarly, as well as other folds that "depend" on a value, or require the fold to terminate before the entire list has been processed.
Anyways, they seem like fairly natural solutions to the problem.