That's at least a 10 hour drive, 599 miles. U.S. 287 has some grades, but nothing major, just wide open prairie. Past Dubois, WY on U.S. 26 you do have Togwotee Pass, but they have much improved this road and it isn't too bad climb/descent. I've driven every permutation of the roads between Sydney and Yellowstone, my preferred route is U.S. 26 to I-25, to Casper, U.S. 26 to Dubois and GTNP to Yellowstone. About a fast as any, low stress, varied and beautiful sights, but it is a long one day haul, I usually break it into two days.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI