Yup. A 172 can cruise at 9500 or 10500 feet without too much trouble. BTDT. It should be able to go even higher, dependent on temperature up there (service ceiling is 14000 density altitude). That's much more altitude than you need for the problem at hand.
It still can be quite turbulent even at those altitudes above the mountains when the wind is blowing strongly. But when you're a mile above the terrain, there is a lot more margin to avoid it. Then, the only things you need to avoid are Va, Vs, and your own barf. A 172 lightly loaded at Va will be slow. 90 KIAS solo with full fuel, down to 80 with empty fuel (check the POH -- there is a little variation).
At the high altitudes, you'll need to climb close to Vy. It's much lower at high altitude than at sea level (but always higher than the sea level Vx). Watch your climb rate and adjust accordingly. You will be cruising at full throttle in the absence of turbulence, so a cruise-speed climb is obviously impossible.