A CPR LTT MADE AN APCH TO THE WRONG ARPT. THEY DID NOT LAND.

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WE WERE CLRED FOR A VISUAL APCH TO THE RAPID CITY ARPT WHILE 15 MI OUT. (APCH INDICATED THE ARPT WAS AT 12 O'CLOCK AND 15 MI. WE OBSERVED AN ARPT AT APPROX THE 11:30 POS AND 15 MI; BELIEVING THIS TO BE RAPID CITY WE ACCEPTED THE CLRNC). THE RWY PATTERN AND ORIENTATION AT BOTH ARPTS ARE SIMILAR; IN ADDITION THERE WERE NO NAVAIDS TO HELP WITH NAV. THE ONLY VOR WHICH WAS OF USE WAS LOCATED 90 DEGS TO OUR FLT PATH AND SEVERAL MI FROM THE ARPT. (BOTH ARPTS ARE INTERSECTED BY THE SAME RADIAL FROM THIS VOR). WHILE ON BASE LEG OF APCH NOTICED THE RWY APPEARED TO BE 'LARGER' THAN I WAS EXPECTING (BY THIS I MEAN IT APPEARED LONGER AND WIDER THAN ANTICIPATED). TURNING FINAL I ASKED THE TWR IF THEY HAD US IN SIGHT. THEY RESPONDED 'NEGATIVE.' AS WE WERE ON FINAL AND CLOSE TO THE ARPT THIS SEEMED TO CONFIRM THE PROBLEM. I BEGAN THE MISSED APCH/GAR. A FEW SECONDS LATER; THE TWR CALLED US IN SIGHT AND INDICATED WE WERE 5 MI N. THE CTLR THEN ADDED 'THAT WAS NOT THE RAPID CITY ARPT.' WE THEN SPOTTED THE RAPID CITY ARPT AND LANDED. CONTRIBUTING FACTORS: 1) EBOUND DIRECTION OF FLT HAD US LOOKING INTO THE SUN; WHICH WAS CLOSE TO THE HORIZON. 2) BOTH PLTS WERE TIRED - - WE LEFT THE HOTEL EARLY AND DID NOT GET MUCH SLEEP. 3) I FAILED TO CONSIDER HOW CLOSE THE OTHER ARPT WAS TO OUR DEST; A SMALL CHANGE IN HDG WAS ALL IT TOOK TO 'CHANGE' OUR DEST. LACKING A VOR OR OTHER NAVAID ON THE FIELD I SHOULD HAVE WAITED UNTIL WE WERE CLOSE ENOUGH TO THE ARPT THAT THERE COULD BE NO DOUBT BEFORE ACCEPTING THE VISUAL.