Reactivity of benzyl halides towards nucleophilic substitution
As you already noted, phenyl bromide is least reactive towards nucleophilic substitution, so any answer suggestion that does not place III last is wrong a priori. The reasoning you gave does not quite add up with me so here is my version: no $\mathrm{S_N2}$ back attack is possible through the phenyl ring since that pathway is blocked by the other ring atoms; and no $\mathrm{S_N1}$ pathway is possible since that would require a very unstable phenyl cation with the empty orbital being of the $\mathrm{sp^2}$ type.

Thankfully, this also immediately tells us that II is penultimate, since no other options exist and the question boils down to whether I or IV is more reactive.

If you were to assume an $\mathrm{S_N2}$ pathway, then indeed I would lose to IV due to increased steric inhibition: the electrophilic carbon is not easily accessible from behind.

However, both reactive carbons are benzylic and the neighbouring phenyl ring is very good at stabilising ionic charges. Considering that you also have a rather weak nucleophile — methanol — it is very likely that this reaction will proceed under $\mathrm{S_N1}$ conditions, so the rate determining step is the formation of the cation. From this point of view, I gives the better intermediate cation than IV because of the additional inductive stabilisation from the methyl group.

Therefore, the correct ordering is $\mathrm{I&gt;IV&gt;II&gt;III}$ or answer C.