There is a photo of that locomotive on page 148 of Greenberg's standard gauge book. He shows it as "...FAOS Special, 0-4-0, c.1915." It looks like the same engine as the #33 c, 1917-1918. There is an NYC decal on the left side of the cab. It's very faint in those two photos but you can see it if you look for it.
If I had that locomotive, I would straighten all of the handrails and accessory mountings, but I wouldn't try to restore the finish. That funny looking thing on top should go.
Does it run. Certainly it can be fixed to run unless the wheels have expanded. But they may be replaceable. If you can get it to run, making the headlight work is easy. The wires probably need new insulation, and you'll probably have to clean the brushes and armature. Some painstaking work but certainly doable. It was no reversible, so that avoids one annoying part of wiring. If I ahd that engine, I would surely get it running.
I don't have that book. Any chance you could snap a pic of the pic and post it here?
The funny looking thing on top is just the loose parts for the loco sitting up there - they could go to where they belong.
I will try to get it spruced up mechanically!