Green’s Dictionary of Slang

(US) to show off, to act like an important person.

1931–4		D. Clemmer Prison Community (1940) 330/1: big shot, [...] vt. To imitate, to simulate the attribute of a superior.
1958		W. Motley Let No Man Write My Epitaph 204: Louie big-shot for the beer, smacking a half-dollar down.
1968		in B. Jackson In the Life (1972) 225: He wanted to go to bed with the girl and was big-shotting.
1977		N.C. Heard When Shadows Fall 161: ‘I done gave you my car, my money, my home and my ass, and I didn’t do it so you could big-shot for some pissy little bitch’.