If it's OK to make a new array it's this:
$result = array();
foreach ( $array as $key => $val )
$result[ $key+1 ] = $val;
If you need reversal in-place, you need to run backwards so you don't stomp on indexes that you need:
for ( $k = count($array) ; $k-- > 0 ; )
$result[ $k+1 ] = $result[ $k ];
unset( $array[0] ); // remove the "zero" element