Does the benzene ring in styrene show a negative inductive effect on the ethenyl group?
This can be explained by looking at the molecular orbitals. The molecular LUMO of benzene (which accepts electrons) is closer to the HOMO of ethylene (which given electrons) as compared to the large gap between HOMO of benzene and LUMO of ethylene.

Surely, the HOMO, LUMO don't exist separately and only exist as a molecular orbital. But the separation makes for easy transfer of electrons from ethylene to benzene unlike the opposite.
Because I have involved pi orbitals into the solution, I believe mesomeric effect is more apt here. To talk about inductive effect we have to involve a more complex idea (group electronegativities). Resources for such data are not easily accessible. I would appreciate if someone can find the group electronegativities of benzene and ethylene and post them here for the analysis of inductive effect as well.
Surely, if present, a soft nucleophile will it will attack the 2-carbon on ethylene side-group.