Green’s Dictionary of Slang

honest, upright, respectable, clean-living.

[1885		Bulletin (Sydney) 25 Apr. 18/1: The latter typical squatter-M.P. has voted with the Government as straight as an arrow all through the Parliament, but never spoke except once, when A. G. Jones, his rival in the silence of silence, said ‘yes.’].
1969		J. Crumley One to Count Cadence (1987) 236: You are a straight arrow square, aren’t you?
1972		O. Hawkins Ghetto Sketches 150: Kwendi shakes his hand again, straight-arrow style this time.
1978		L.K. Truscott IV Dress Gray (1979) 297: He becomes a regular gray-hog [...] That’s like a super-straight cadet [...] Gottlieb digs in, and he’s straight-arrow.
1989		M. Amis London Fields 87: The thing was, the thing was . . . he was straight arrow.
1990		(con. early 1950s) J. Ellroy L.A. Confidential 40: Straight-arrow Eddie, war hero with a poker up his ass.
2005		E. White My Lives 233: Just a straight-arrow glance of friendly recognition.
2023		(con. 1962) J. Ellroy Enchanters 191: ‘[H]e’s a straight-arrow guy’.