Work vehicle sprayer system and method with self-cleaning filter apparatus

A self-cleaning filter apparatus includes a filter unit and a plunger unit. The filter unit includes a screen device; and a filter unit housing that houses the screen device. The plunger unit includes a rod end housing coupled to the filter unit housing; a plunger rod with a first plunger rod end positioned within the rod end housing and a second plunger rod end within the filter unit housing; and a plunger assembly secured to the second plunger rod end. During a first filter cleaning event segment, the plunger rod is configured to translate between a nominal position in which the plunger assembly is proximate the first screen device end and a cleaning position in which the plunger assembly is proximate the second screen device end. The plunger rod is configured to dislodge, at least in part, debris from within the screen device during at least the first filter cleaning event segment.