It's complicated. The league will definitely submit their punishment before the start of the season. If Watson sues the league because of the punishment severity, then he would be allowed to play Week 1, up until whenever a verdict would be reached the trial. That could be 8 weeks, it could be a year. Happened to Brady with deflategate, he got to play in 2015 because of the trial, hence why the suspension was during the 2016 season.
The thing is, it might not be smart for him to sue the league, if he truly values his money as it might end up costing him more. Judge Robinson findings concluded that he is absolutely guilty of breaking the NFL's code of conduct, but the reason she didn't suspend him for more games was because the NFL hadn't set a stern precedent on punishments, so they cant give him more without fair warning. Which quite frankly is stupid, but whatever. Either way, since she found him guilty, the federal court will most likely side with the NFL on the issue. If that's the case, if Watson draws this out into the 23-24 season, he could lose even more change due to the way the Browns structured his contract to help mitigate financial loss for a suspension this season. I think he's only set to make $10M this year, where as next year he'll be making $46M, which is quite a bit of dough out the door, assuming there isn't another restructure.