First let's make sure we've answered Halls' questions, Nusc.
You have: \langle f, g \rangle = \int_0^1 f(x) \overline{g(x)} \, dx
and: Vf = \int_0^x f(s) \, ds
Now, you want: \langle Vf, g \rangle = \langle f, V^\ast g \rangle. The task being to find V^\ast. You have actually been given a V^\ast, so are only required to check it.
The left-hand side of this equality translates to
\int_0^1 (Vf)(x) \overline{g(x)} dx = \int_0^1 \left( \int_0^x f(s) ds\right) \overline{g(x)} dx
We may write this as a double integral: \int_0^1 \int_0^x f(s) \overline{g(x)} \, ds \, dx \; \; \; (\ast)
Noting that the required right-hand side is of the form
\langle f, V^\ast g \rangle = \int_0^1 f(t) \overline{(V^\ast g)(t)} dt
(where t is an arbitrary parameter: above we used 'x')
we see from (*) that finding V^\ast just requires one to switch the order of integration: take t=s; we want ds dx to become dx ds.
There is no "integration by parts" required. This work comes solely under "double integrals".