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Converting the polygons to rasters (using the area as the raster value) and then extracting the raster value to the points is the obvious "easy" process.
Doing an intersect between the polygons and the points would "extract" the values into another point layer that you could join to and copy value to your points.
Haven't see a direct extract from the polygon to the point. Maybe someone will reveal that via this thread.
I tried spatial join and couldn't get it to work exactly how I wanted it to. I found the parameters of the tool a little confusing. However, converting the polygons to points and extracting them worked fine for me.