We(Japanese) generally don't say like that. But we say 'シャンハイじんshanhaijin（上海人person from Shanghai）'. I think it came from Chinese way. And we also say 'おおさかじんoosakajin（大阪人person from Osaka）' and 'かんさいじんkansaijin（関西人person from Kansai）'. I don't know whether it came from Chinese way or not. We don't say 'かんとうじん kantoojin（関東人person from Kanto）' or 'とうきょうじんtookyoojin（東京人person from Tokyo）' or 'ほっかいどうじんhokkaidoojin（北海道人person from Hokkaido）person from Hokkaido' etc. We say 'とうきょうしゅっしんtookyoosyusshin （東京出身person from Tokyo）' and 'ほっかいどうしゅっしん' etc. We can say 'おおさかふみんoosakafumin（大阪府民Citizen living in Osaka）', 'とうきょうとみんtookyootomin（東京都民Citizen living in Tokyo）', 'ほっかいどうみんhokkaidoomin（北海道民Citizen living in Hokkaido）' etc about a person living there now even though his/her original hometown is not there. I don't know the reason why we say only 'かんさいじん' and 'おおさかじん' so often. But I can imagine that they were thought very different to Yedo(えど・江戸)＝Tokyo people.

I am pretty sure 大阪人 is fine! I googled that phrase and many results came up for both the romaji and the kanji. When I was in high school in Kagoshima Prefecture, we called ourselves 鹿児島人, but I think Placeの人 seems a little more polite/proper.