Really the best route is MT 308 to WY 72. It's 63 mi. I'm not sure how you'd do it using I-90, it goes way out of the way.
From Cody, go E on US 14 to Greybull, S on US 20 to Worland, US 16 to Buffalo. You will go through the Bighorn Mtns, but it's a much better road than US 14. You will go over Powder River Pass, but it's well maintained, passing lanes on upgrades, and much to see.
As on any climbing, pay attention to temperature and downshift to keep engine in it's best torque range. Don't worry MPH, a slightly higher RPM means cooler engine and less lugging. Stop and take pictures, stay relaxed!
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI