This is a reference to trying to walk straight (walk a line) and pass a field sobriety test, while drunk or intoxicated, in front of a Law Enforcement Officer.
A man passes a field sobriety test while drunk by walking in a straight line. By walking straight, he is released and free to go, even though he broke the law. He fooled the cop and does not have to go to jail.
Fogey/fogy /fougi/ sl. (early 18C+, orig. Scot) old-fashioned, stuck-in-the mud.
Person with old fashioned ideas which he is unwilling to change: Come to the disco and stop being such an old fogey!
You think me an old fogeyand an old tory, his thoughtful voice said. I saw three generations since O’Connel’s time. I remember the famine. Do you know that the orange lodges agitated for repeal of the union twenty years before O’Connel did or before the prelates of your communion denounced him as a demagogue? You fenians forget some things. (James Joyce, Ulysses. Penguin Books,1992. p. 38)