You can build a regular expression from your string. If you just want the characters appear in the same order, you can place .* between each two characters. The regular expression for "filswjs" will then be f.*i.*l.*s.*w.*j.*s, which means "f, than anything, then i, then anything, then l, ...". Then just grep the arrays for matches:

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

my @array1 = qw(fil01_sw100056_02.04.js mn_sw100868_15.17.js
                roof_app01_sw12345_26.02.hex roof_dir01_sw101481_01.32
+.dir);

my @array2 = qw(fil01_sw100056_02.03.js mn_sw100868_15.16.js
                roof_app01_sw101483_25.05.hex roof_dir01_sw101481_01.2
+5.dir);

my $string = 'filswjs';

(my $regex = $string) =~ s/(.)(?=.)/$1.*/g;
say for grep /$regex/, @array1, @array2;
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