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Step-by-step build — 5 steps. Also available as a print sheet.
Stand the three dividers on the bottom shelf and fasten them — pocket holes into the shelf, or countersunk screws and glue up from underneath. Space them evenly so the cubbies come out the same width.
Attach the 1x2s inside the dividers to strengthen the assembly and keep the cubby faces consistent. If you don't have a pocket-hole jig, you can skip ahead and add these after the top is on, fastening them with 1-1/4" finish nails and glue up into the top instead.
Set the top shelf on the dividers and stiffeners. Fasten to the dividers, and use 1-1/4" finish nails and glue into the 1x2s. Keep the assembly square as you go — a racked cart binds when it rolls.
Nail the 1/4" plywood back on with 1-1/4" finish nails. The back squares the whole cart and stops it from racking, so tack it on all four edges.
Screw the four casters to the bottom, then roll the cart into the bench or hutch opening to check the fit and clearance. It should slide in and out freely with the top just clearing the underside of the bench. Adjust caster height if it scrapes or sits too low.