A standard way to catch errors is to use die (or Carp), in combination with eval. But since I prefer Try::Tiny, I'm going to use that.
For example:

#!/usr/bin/perl

use strict;
use warnings;
use Try::Tiny;

$main::errorA = 1;
$main::errorB = 1;

sub methodA {
    if ($main::errorA) {
        die "Something went wrong in methodA";
    }
}

sub methodB {
    my ($handle) = @_;
    &$handle();
    if ($main::errorB) {
        die "Something went wrong in methodB";
    }
} ## end sub methodB

my $rowHandle = \&methodA;

try {
    methodB($rowHandle);
}
catch {
    print ("ERROR: " . $_);
}
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This results in:
ERROR: Something went wrong in methodA at ./monks24.pl line 12. (monks24.pl being the name of the executable).
This is because errorA is checked before errorB.
If you'd set errorA to 0 (meaning all is well in methodA), you'd get this:
ERROR: Something went wrong in methodB at ./monks24.pl line 20.