If I understand your requirements correctly, the following should work. (It works with your example strings as you have provided.)

use strict;

my @strings = qw(aaa_gjh1_dfgdf_0009 aaa_gjh_0000 aaa_fdsfs_000 aaa_sd
+f_jdsh_01111);

foreach my $test (@strings) {
    if($test =~ /^aaa_[a-z0-9_]{1,23}[a-z_][0-9]{4,4}$/){
        print "Match---\n";
    }
    else{
        print "No Match... :(\n";
    }
}
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