He lost his job ___no fault of his own.

选A
through no fault of his own的意思为“并不是由于他本人的过错”,其中的through是介词,表示“由于”的意思,相当于because of和as a result.例如：
The accident happened through no fault of yours.这场事故的发生不是你的错.
We got lost through not knowing the way.由于不认路我们迷路了.
It was through you that we were late.都是因为你我们迟到了.
through的另一个意思是“经由……”,“以……”,“依靠……”,表示工具,方法,手段,相当于by means of.例如：
He learned English through the radio.他通过收音机学英语.
She got the job through his uncle.她靠她叔叔的影响谋到了这份工作.
by通常是“通过” “由于” “凭借” with 表示“和...” over 是“过了...” 由语义得出选A