I'm assuming that your polyline feature to which you want the distance is in a separate feature class than your road centerlines are. If that's not the case, let me know.
You could try creating point at every intersection of the roads with your polyline feature (You could use the Intersect tool for this, Intersect—Help | ArcGIS Desktop ). The create an OD Cost Matrix Analysis layer (OD cost matrix analysis layer—ArcGIS Pro | ArcGIS Desktop ) and load the intersection points in as your destinations and the other points in as origins. Then solve for the 1 closest destination.
Note, using the OD Cost Matrix analysis will return straight line geometry output, but the distances reported will be for the driving distance. If you want the actual geometry, then use the Closest Facility analysis (Closest facility analysis layer—ArcGIS Pro | ArcGIS Desktop ) and load the intersection points as facilities and the other points as the incidents.
Hope this helps!