I think a combination of factors will lead to low production numbers for Hillis in the season opener.
Cincinnati will have a poor offense this year. There's no way around that. At the same time, however, their run defense was excellent in the preseason and was pretty good last year, too.
The Browns won't have much trouble stopping the Bengals. That will encourage Shurmur to use the game as an additional "developmental" scrimmage (He's still new at this, after all). He'll want to see as many of his players as possible. Additionally, the offense that they fielded in the preseason seems geared towards spreading the ball around, with more passing than the Browns had last season.
Hillis will be effective, but I'm thinking a 65yd rushing day with 3 catches for 45yd is the most we can hope for in Game 1. And the Browns will win easily.