Lifting a congruence $f(X) \equiv gh (\textrm{mod} \ p)$ to a higher power moduli
One way to raise solutions to higher powers of the modulus is Hensel's Lemma.

First, you need to express your element as a root of a function: $g(u)= hu - (X-1)$
Find the solution modulo 2, which you already did.
Use the Lemma to raise this solution to higher powers of 2. Maybe it is useful to do this in several steps. Find the solutions mod 4,8,16 and then 32.