The filter won't do anything for reflections internal to the lens. Using a lens shade will help as will not shooting into strong light.

Softness could come from many sources, anything from a characteristic of the lens, to some flaw like not being correctly fitted to the shutter or internal cloudiness from lubricants or 70-ish years of temperature and humidity cycling.

A lens doesn't have to be coated to be sharp, but it helps.
Mostly, so I understand, coatings improve contrast which does help with apparent sharpness.