City were the first Manchester club to have the name ‘Manchester’.
City were the first Manchester club to win a trophy of any kind with the 1898-99 Second Division title.
City were the first Manchester club to win a major trophy with the 1904 FA Cup.
City were the first Manchester club to score 6 in a derby (1926).
City were the first Manchester club to get an attendance above 80,000.
City were the first Manchester club to play European fixtures when we were invited on our Champions tour of Germany in 1938:
https://www.google.co.uk/amp/s/www.independent.co.uk/sport/football/european/when-manchester-city-played-berlin-and-foiled-nazi-propaganda-machine-a6672906.html?amp
City were the first Manchester club to be relegated as champions and the league’s top scorers.
(first club of all in England… not all history has to be positive, haha!)
City were the first Manchester club to win a double with the 1970 EFL Cup and European Cup Winners’ Cup.
(that was also the first time an English club won a double that included a European trophy)
City were the first Manchester club to get 100 points in the league
(first club of all in England).
City were the first Manchester club to win all four domestic trophies in one season
(first club of all in England).
And like you say, City were the first Manchester club to win a Manchester derby FA Cup final.