In the 1950's how were radio-astrometric positions with portable dishes so precise they could be assigned to their dim optical counterparts (Quasars)?
In many cases in astronomy, there’s a difference in finding the position of something precisely vs. seeing detailed structure in that object.  In the former case, what you need to do is to find the centroid of the image.  So even with relatively coarse spatial resolution, you can often get very precise positions because you can centroid the images very well.  (EDIT: However, this may not apply to early radio observations because single-dish radio telescopes don’t form images.)
Specifically for 3C273, the first quasar identified, Wikipedia says that its position was refined with lunar occultation. This technique involves observing a source as the Moon approaches it and determining the precise moment when it disappears behind the edge of the Moon. Here you need to measure time precisely and to know the Moon's path very well.
Also see this answer to Has radio astronomy ever been done on objects that appear very close to the Moon? Is this avoided?
The first radio source to have a star-like optical counterpart associated with it was 3C84, and that was done by measuring the radio position precisely using interferometry, i.e. combining signals from multiple telescopes.
So I think that this was not in fact possible with single-dish radio telescopes in the 1950s, and required more advanced techniques that came a bit later.