Using polygons to apply unique value to all raster cells within those polygons?

You can use a conditional statement. The issue with previous recommendations is that when you rasterize your polygons (which is necessary) the background, that does not contain polygons, will be NoData resulting in corresponding areas in the output also being NoData. You will need to set your analysis extent to your original raster and then set a background value (i.e., 0) to the rasterized polygon raster using SetNull. Once you have done this a conditional statement in the raster calculator will look something like this.

Con("praster" > 0, "praster", "OrgRaster")

"praster" is your rasterized ploygon feature class, with a background value of 0, and "OrgRaster" is the raster you wish to modify. This statement is saying that if praster is greater than 0 then assign values from praster else assign values from OrgRaster.

You can rasterize (ArcToolbox > Conversion Tools > To Raster) your polygons, and then merge the two rasters with Spatial Analyst Tools > Math > Logical > Over.

While rasterizing polygons, in the Polygon To Raster dialog window, you need to use Environments -> Processing Extent Snap Raster option, to get the cells correspondent to your initial raster.