You have the penultimate bracket in the wrong place, and you should line your ranges up:
=INDEX('[World Population.xlsx]Sheet1'!$B$3:$BX$291,MATCH($B4,'[World Population.xlsx]Sheet1'!$A$3:$A$291,0),MATCH("1999", '[World Population.xlsx]Sheet1'!$B$2:$BX$2,0))
or if the year is stored as a number, use:
=INDEX('[World Population.xlsx]Sheet1'!$B$3:$BX$291,MATCH($B4,'[World Population.xlsx]Sheet1'!$A$3:$A$291,0),MATCH(1999, '[World Population.xlsx]Sheet1'!$B$2:$BX$2,0))
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