If fans turn or if Arteta stay however long should be the same criteria as for any other manager. A combination of results, tactical development and how he's utilizing the squad at his hands. Of course patience is needed, but look what indecisiveness and "patience" got Arsenal into with Emery, when he really should've been fired at the end of last season.
I don't think you can go into a project and just state "he will need so and so many years". You got to continuously evaluate what's happening in terms of results, set out goals, tactical development and squad development/utilization. E.g. if the club and fans - and I think in the case of Arteta becoming manager and the club very likely not getting and international fooball next season - goals and league finish expectatiosn should be lowered as a proper rebuild will be at hands: Leaving senior players, a rookie manager, very many young players, probably not too many top players joining.
So e.g. if there's a clear plan at hand, a clear tactical handwriting which is developing, the squad is used in proper fashion, but then there's a string of bad results - that's when you want patience in such a situation. If these things don't happen you'd best pull the trigger even when results are still kind of alright. And if on top of that results are bad, you pull the trigger no matter if a manager has been at the club 1 months, 6 months or 2 years.
Great post, 100% agreed..