How did the first manned Moon mission spacesuits block the deadly radiation of both Van Allen belts and direct sunlight?
The Van Allen belt radiation was primarily shielded by the hull of the spacecraft, not the space suits. The electrons were mostly absorbed by the aluminium. Of the protons, however, some would penetrate the hull, and the flight path was chosen in such a way as to avoid the most intense radiation.
On the Moon, there's no Van Allen belt radiation, so only solar and cosmic radiation had to be dealt with, plus a small dose of secondary radiation produced by absorption processes in the regolith. A space suit cannot protect against all of this radiation, but it was good enough to not be of major concern.
There was however a constant danger of solar coronal mass ejections while a mission was flown, which would create huge amounts of high energy radiation. This was a risk that simply was accepted. This is one of the major obstacles for a manned mission to Mars. While the probability of a CME during a one week mission is acceptably low, it's almost a certainty for a one year mission.