Is it possible to combine rod brakes with a coaster brake?
You won't be able to simply bolt on a coaster brake. Coaster brakes are typically built into a single speed or 2 or 3 speed internally geared hub.

To fit a coaster brake you will have to replace the rear wheel hub, which obviously means a wheel rebuild, or whole new wheel with a matching rim.

Whether a modern coaster brake hub would fit in your frame depends on the dropout spacing of your frame. I guessing that single speed or 3 speed coaster brake hubs come in narrower widths as used on older bikes, likely around 120mm.

If your bike already has a internally geared hub then there are extra complications, if you replace it with a modern internally geared hub you need to fit a shifter and necessary cabling.