"My code works but I am not sure if I am over doing it or if there would be a better and more straight forward way to accomplish the same thing more efficiently."

Using the core module Time::Piece, I think you'd agree this is more straightforward:

#!/usr/bin/env perl -l

use strict;
use warnings;

use Time::Piece;

my $this_year = localtime->year;
my @filenames = ('file_name_' . ($this_year - 1) . 12);
push @filenames, "file_name_$this_year$_" for '01' .. '11';

print for @filenames;
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Output:

file_name_201212
file_name_201301
file_name_201302
file_name_201303
file_name_201304
file_name_201305
file_name_201306
file_name_201307
file_name_201308
file_name_201309
file_name_201310
file_name_201311
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Whether it's "better" is somewhat subjective; however, I think I'd prefer to maintain the three lines I use to generate the array of filenames to all the code you posted.

Use Benchmark (also a core module) to compare the running times.