I replaced the 8 speed crank on my 2013 Trek Marlin with a 9 speed XTR crank and it runs just fine with the 8speed chain and derailer. So I don’t think you will have any problems.
9 speed cranksets have narrower chainrings, as their chains are narrower. So using a wider (8-speed) chain will work just fine on a 9-speed crankset, which is what you want to do. (But note the opposite is not true: a 9-speed chain will [i:3assywh3]not [/i:3assywh3]fit on a 8-speed crank, as the chainrings are just too wide for the narrower chain.]
PS It’s been mentioned, but that Amazon crank you listed is not 9-speed anyways.