Date::Calc is great for this sort of thing:

#!/usr/bin/perl -w

use strict;
use Date::Calc qw(Delta_Days);

my $curdiff = 1e10;
my $nextrow = 0;

my $array = [ { date => '2001-02-01', more => 1 },
              { date => '2001-03-31' },
              { date => '2001-04-15' }
            ];

my $today = '2001-03-28';

my ($nowy,$nowm,$nowd) = split(/-/,$today);
foreach (my $rownum = 0; $rownum < scalar(@{$array}); $rownum++) {
  my ($nexty,$nextm,$nextd) = split(/\-/,${$array->[$rownum]}{'date'})
+;
  my $delta = Delta_Days($nowy,$nowm,$nowd,$nexty,$nextm,$nextd);
  next if $delta < 1;
  if ($delta < $curdiff) {
    $nextrow = $rownum;
    $curdiff = $delta;
  }
}

# for insert mark
$array->[$nextrow]->{'isnext'} = 1;
# for array ref
my $arrayref = $array->[$nextrow];
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Of course, this could be optimized more fully, but it should be enough to get you started.