It's not immediately apparent how to fix this. Push-and-turn doesn't cut it. Any ideas?

It's in two pieces. You'd think you would just screw them together.

With no better data or images- Apply film of grease to insides of silver barrel. Insert the plunger and thread it's plastic end cap into the end of the barrel, thread the cap into the barrel. Open the chuck's compression lever, it's in the closed position currently. Stroke the pump handle a few times to spread the grease along the barrel. It's now ready for use.

This pump has no "frame fit/high pressure pumping" feature in the handle, unlike the HPX version. Storing or leaving the chuck's lever in the compressed position will negatively affect the valve sealing gromet. Always open the lever so it lays flat along the barrel when not in use. Andy (who has had quite a few of these pumps back in the 1980s)

Guessing at what the problem is from your incomplete description:

The fitting that screws into the pump barrel is not visible in the photo because it's sitting in the handle, above the spring. Move the spring down from the handle and then fish the threaded fitting out of the handle. You should then have no trouble screwing the fitting into the pump barrel, making the pump operational again.