First off, you have an error in February's conditional. Secondly, you could take advantage of perl's if-it-looks-like-a-number-it-is and do this:

my @months = qw(January February March April May June July August Sept
+ember October November December);
print $months[$realmonth-1];
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Thirdly, since you're dealing with dates, you should probably be using one of the DateTime modules or at least POSIX::strftime.

Update: Fixed the off-by-one error in my array index.