Try:
Well, we know it will succedd as long as $g'(1) < 1$. Notice that
$$ g'(x) = \frac{ (\pi/2) x^3 \cos (\pi x/2) - 3 x^2 \sin (\pi x /2) }{x^6} $$
and $|g'(1)| = 3 $. Thus, the fixed point iteration will fail to spot the root.
To answer b), using Mewton' method $f(x) = x^4 - \sin ( \pi x ) $
and so $g(x_n) = x_n - \frac{ f(x_n ) }{f'(x_n) } $.
is this correct solution?