The wiki suggests it's possible but not guaranteed. Here's the situation:
What favors Watson staying in: Coach Todd Monken explicitly said after naming Watson the starter that he is "not anticipating" week-to-week changes at QB, indicating Watson has a stable path as the primary starter (from the Shedeur Sanders page).
What favors a benching if he struggles: The Sanders page also states that Sanders will "have opportunities to play if Watson struggles early or faces injury." Sanders was close enough in the competition that Monken needed the full preseason to decide — and the Browns' organizational dysfunction, Watson's poor Cleveland track record (just 19 TDs in 19 starts), and his toxic relationship with the home crowd all create pressure (from the Deshaun Watson page).
On The Fantasy Headliners, the discussion around the Browns' QB room highlighted that Cleveland spent two draft picks on wide receivers and focused on the offensive line — setting up whoever plays QB, but the question lingers over whether Watson is the guy.
Net take: If Watson is really bad for a month, the leash is probably shorter than Monken is letting on — Sanders is right there, and the front office invested a pick in him. But if Watson is merely mediocre, he'll likely get the full season to prove it.