If you want native resolution, you're going to need to get the correct cables.
HDMI max resolution is 1920x1080 (reckon it can do 1920x1200 on a 16X10 display)
DVI-D max resolution is 1920x1200 (technically, but sometimes can get higher - rare, but not to match that monitor)
VGA max resolution is about where you are at.
To get the monitor's native resolution, you're going to need either a displayport or a dual link DVI.
With your late 2012 Mac mini, you would need a 'mini displayport' to 'displayport' cable.
If Amazon is making that in their Amazon basics, have found their cables to be pretty good at a good price.
There is also monoprice.com.