LGT FLT CREW CONDUCTED VOR APCH TO WRONG RWY RESULTING IN OVERSHOOT AND SUBSEQUENT GAR.

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ENRTE JFK; BEFORE DSNDING; CALLED UP JFK ATIS INFO AND CAPT BRIEFED VOR 13L APCH. AFTER CONTACTING NEW YORK APCH AT CAMRN HE INFORMED US THAT NEW ATIS INFO WAS CURRENT. COMPLIED WITH VECTORS AND REQUESTED NEW INFO ON THE ACAR'S. RECEIVED NEW INFO AND WHILE READING IT; APCH TOLD US TO KEEP OUR SPD UP AND PROCEED DIRECT TO THE IAF. I NOTED THAT THE WX WAS DIFFERENT AND RWY 4/22 WAS CLOSED BUT FAILED TO NOTICE RWY 13L; THE NORMAL LNDG RWY; WAS CLOSED. COMPLETED THE CHKLIST AND TOLD BY APCH THAT WE WERE #1 AND TO KEEP THE SPD UP AS MUCH AS POSSIBLE. HE THEN CLRED US FOR THE VOR 13R APCH. WE ACKNOWLEDGED THE CLRNC BUT BOTH OF US HAD THE VOR 13L APCH FIRMLY PLANTED IN OUR MINDS. MONITORING RAW DATA; I COMMUNICATED THAT WE WERE L OF THE COURSE AND PASSING CRI WE DSNDED TO 1000 FT. PASSING THE FIRST SET OF SEQUENCING FLASHERS; I NOTED THAT THINGS WERE NOT RIGHT SINCE THE MIDDLE SEQUENCERS WERE NOT FLASHING BUT THE RABBITS FOR RWY 13L WERE FLASHING. JUST AS I WAS REALIZING OUR MISTAKE THE TWR CALLED TO REMIND US THAT RWY 13L WAS CLOSED. WE STARTED TO TURN TOWARDS THE R SIDE BUT IT REQUIRED A TIGHT TURN AND WE WERE HIGH ON THE APCH. WE ELECTED TO ASK FOR A GAR AND COMPLETED A NORMAL APCH TO RWY 13R. FATIGUE WAS DEFINITELY AN ISSUE. I WAS AWAKE SINCE XD30 AND ALTHOUGH I ATTEMPTED TO NAP; I WAS UNABLE TO FALL ASLEEP. WE WERE ALSO PERFORMING 3 DAYS ON 2 OFF ON A SEQUENCE THAT GOES FROM LATE ARRS TO EARLY WAKE-UPS IN 3 DAYS.