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I think there were two styles of headsets in play for that time period. One, like Jason's where there a cup is in the top of the head tube, and you thread down the race. And the other, where the race is in the head tube, and the cup threads down.
From the pic it kinda looks like your '51 has the type where the cup threads down. I was recently looking for one of that style and had a harder time then just the 'normal' one (like Jason's. I say normal because I personally have seen more like that).
Or, if you want a Schwinn one of vintage, they can be found on eBay (I know, I know) for not too bad of pricing.
I'm running this one on two of my bikes-- http://porkchopbmx.com/vp-h755-bmx-bicycle-headset-1-threaded-w-32-5mm-cups-chrome/
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