Which 1km data set are you using for your input? Natural Earth?

What is it that you plan to do with the reprojected map?

https://gis.stackexchange.com/questi...ar-to-mercator might be of use to you as it shows how to take a whole-world raster image (ETOPO1 GeoTIFF) in the Equirectangular projection (epsg:4326) and convert it to a cropped GeoTIFF in the Mercator (EPSG:3857).
https://gdal.org is the home of the GDAL software (it has useful reference material). https://www.qgis.org/en/site/ is its GUI friend QGIS.
https://www.gisinternals.com/release.php is a good Windows GDAL distribution that you need to actually make the command-line things run. I recommend the x64 distribution if possible because there aren't really too many 32-bit OS versions left out there these days.
Your best bet is likely to be QGIS if you're not familiar with command-line tools, but then you get to learn QGIS as opposed to plunking down some commands for a one-shot activity.

The program should have no problems handling 43200x21600, but you'll probably need to handle the vertical extents because the north and south poles on Mercator are at infinity in its pure form. The stack exchange link above shows how to crop the input to a smaller shape.