Here's a concise version that uses a hash, a hash slice, and substitution.

my %months;
@months{1..12} = qw/January February March April May
                    June July August September October
                    November December/;
my $numeric_month = "01"; # Test case.
my $string_month;
$numeric_month =~ s/0(\d)/$1/;
if ( exists( $months{$numeric_month} ) ) {
    $string_month = $months{$numeric_month}
} else {
    die "Invalid month format.\n";
}
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