The protagonist here is saying that he cannot prove that the reality that he has used to - his normal life - is more real than this other reality of the marble city.
He says "...for by what means can I prove the greater reality of that other life in the house of stone and brick...",meaning that he cannot prove that this marble city is less real than "...that other life in the house of stone and brick south of the sinister swamp and the cemetery on the low hillock, where the Pole Star peeps into my north window each night.."
We see in the beginning of the story that he's used to the Pole Star shining in his window:
Into the north window of my chamber glows the Pole Star with uncanny light. All through the long hellish hours of blackness it shines there.
And right before he says this quote that you're confused about, the story says that our protagonist now desires to actually walk through that marble city that he sees. So since the reality that he's concerned with at the moment is that of the marble city, that makes the other reality normal life - and gives a description of that normal life, describing his house and the star that always shines through his window.
Basically, he cannot prove that normal life is "more real" than this marble city.
