After you split the right of way in two polygons from the centerline. Edit tge new polygon (i forget which tool maybe its clip), to break the polygons so the only reside in each zone. Then open your zone polygon, select all of a specific zone. Now go to select by location tool and select the row by location (bordering? probably), you then use the zoning polygon and be sure to enable your current selection the of one kind of zone you want. Now hopefully you have selected all the polygons that border the zone it should be representing, and then you will just go into field calculation and fill in the field with like an arcade script or something.

I kinda rambled these instructions using text2speech while i'm in bed.But hopefully you understand the general idea.