The paperback anthology was called something like "12 by 13" and one of the stories involved a man who is told to stay in his apartment so that he can survive while earth is destroyed. He peeks outside and sees a sort of fog which he is told is the ichor of the gods. For some reason he is spared destruction.
You may be thinking of Robert A. Heinlein's novella "The Unpleasant Profession of Jonathan Hoag" (from Unknown Worlds, October 1942, available at the Internet Archive), although it doesn't match your description perfectly. There is a Wikipedia article about it.
The paperback anthology was called something like "12 by 13"
The novella was reprinted in a collection of six Heinlein fantasies, sometimes titled The Unpleasant Profession of Jonathan Hoag after the lead story, but there was a Pyramid paperback edition titled 6 X H which was reprinted (with various covers) every year or two from 1961 to 1975; could that be what you were thinking of? The other stories in the collection are "The Man Who Traveled in Elephants", "'All You Zombies'", "They", "Our Fair City", and "'And He Built a Crooked House'".
and one of the stories involved a man who is told to stay in his apartment so that he can survive while earth is destroyed.
The man is not in his apartment, he's in his car, accompanied by his wife, and he was advised to keep his windows closed. The earth is not exactly being destroyed, but our reality is undergoing repairs:
"Where are you going?"
"Back to myself. After I leave, you must do this: Get into your car and drive at once, south, through the city. Under no circumstances open a window of your car until you are miles away from the city."
"Why? I don't like this."
"Nevertheless, do it. There will be certain—changes, readjustments going on."
"What do you mean?"
"I told you, did I not, that the Sons of the Bird are being dealt with? They, and all their works."
He peeks outside and sees a sort of fog
She complied, then gave a sharp intake of breath and swallowed a scream. He did not scream, but he wanted to.
Outside the open window was no sunlight, no cops, no kids—nothing. Nothing but a gray and formless mist, pulsing slowly as with inchoate life. They could see nothing of the city through it, not because it was too dense but because it was—empty. no sound came out of it; no movement showed in it.
It merged with the frame of the window and began to drift inside. Randall shouted, "Roll up the window!" She tried to obey, but her hands were nerveless; he reached across her and cranked it up himself, jamming it hard into its seat.
The sunny scene was restored; through the glass they saw the patrolman, the boisterous game, the sidewalk, and the city beyond. Cynthia put a hand on his arm. "Drive on, Teddy!"
which he is told is the ichor of the gods.
There is mention of ichor in Heinlein's story, but it's not the fog, it's the reddish-brown stuff under the title character's fingernails:
"The stuff under my nails has little to do with the story. It served its purpose, which was to make fearful the Sons of the Bird. They knew what it was."
"But what was it?"
"The ichor of the Sons—planted there by my other persona."