Is 1,2-dihydronaphthalene "partially aromatic"?
Aromaticity is a quite difficult concept but in this case we can answer the question by simply looking at similar compounds and experimental data.

What you got here is a styrol derivative, the pi-system (which we need to look at if we are talking about aromaticity) is the same. So if this compound is not aromatic then styrol shouldn't be either.

But styrol is in fact aromatic (or better: the benzene ring in styrol is). It reacts like an aromatic compound should, it got the right shift in NMR, the bond lengths are in accordance with aromaticity and so is the UV/VIS spectrum. 

So a conjugated double bond clearly doesn't make a benzene (significantly) less aromatic.