Unfortunately anything less than the league and the cup in his first season, and the pressure will ramp up, mostly from external sources and impatient fans, regardless of how good the football is or not. I'm not sure how realistic it is to expect that in a first season, and but there is an expectation here at this club to compete.
Whoever comes in will be made to understand that, we are capable of the title technically and physically, the league was imo pretty poor this past season, and I see no evidence that arsenal are about to create some kind of legacy - lowest number of goals scored by the champions since Leicester, and just a rule change away from having 40-50% of their expected goals disallowed.
Pellegrini got the league in his first season in England and was universally ignored for the success by most of the media. Whilst us as fans and the club will give whoever it is the time to build and create that team and not expect everything at once, barring a finish outside of the European places which I feel would lead to a dismissal, there has to be a push for silverware immediately also.