Probably the easiest this would be to create a constant raster with a value of 0 then with the same cell size as your existing raster layer but the extent of your rectangle. Then you could use any number of raster calculator functions to create a new raster with both rasters as inputs such as the Plus Tool to add them together or use a Over function to copy over the cells from the original where they already exist and apply 0 values to areas where they did not exist before(Unless you have some negative values in your original raster, please look at the raster math toolbox to find the tool to suit your need). Please take into account what you are using for a snap raster environment, since likely your cells will not perfectly overlap and you will want to shift the constant raster to your existing data, instead of the reverse.
Constant Raster - https://desktop.arcgis.com/en/arcmap/10.4/tools/spatial-analyst-toolbox/create-constant-raster.htm
Over - https://desktop.arcgis.com/en/arcmap/10.4/tools/spatial-analyst-toolbox/over.htm
Plus - https://desktop.arcgis.com/en/arcmap/10.4/tools/spatial-analyst-toolbox/plus.htm
Snap Raster Environment - https://desktop.arcgis.com/en/arcmap/10.4/tools/environments/snap-raster.htm