First off, there are some perl modules that do all of this stuff for you. Date::Calc and Date::Manip are both good starts. (I'm sure someone will have posted links before I finish typing this.)

But, in the interest of learning a programming trick or two, though, consider doing your month math something more like:

my @months = qw(JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC);

sub change_num_to_month {
     my $num_month = shift;
     return $months[$num_month - 1];
}

sub add_month {
    my $month = shift;
    my $month_num = change_month_to_num($month);
    $month_num++;
    $month_num = 1 if $month_num > 12;
    return change_num_to_month($month_num);
}

# And very similarly for subtract month
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Getting the reverse lookup is slightly trickier. I'd probably convert the array into a hash, something like: (I'm sure you can do better with a map, but let's go for readibilty first.)

my %month_nums;
for (my $i = 0; $i <= $#months; $i++) {
   $month_nums{$month[$i]} = $i;
}

# This creates a hash of 'Jan' => 0, 'Feb' => 1, etc.

sub change_month_to_num {
    my $month = shift;
    return $month_nums{$month} + 1;
}
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One has to be careful about the array index starting at 0, if you want your month nums to range from 1 to 12, but that's the only really tricky bit.