If something feels redundant, that's probably a good reason to look at putting it in a subroutine. I would do something like this:

sub handle_options
{
    my ($hash, $options, $subref) = @_;

    foreach my $try (@$options)
    {
        next unless exists $hash->{$try};
        $subref->( $try );
        return;
    }

    die;
}

eval {
    handle_options( \%hash, \@options, \&do_something );
}; if {$@) {
    log_failure();
}
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(Yes, I am really returning undef and die'ing with no string. The key is die/not-die, determining whether or not to trigger the if ($@) { ... } block.)

The really neat thing about this method is that you can handle multiple sets of options in the same eval, so long as they all use the same algorithm.

eval {
    handle_options( \%hash1, \@options1, \&do_something1 );
    handle_options( \%hash2, \@options2, \&do_something2 );
    handle_options( \%hash3, \@options3, \&do_something3 );
}; if ($@) {
    log_failure();
}
[download]

Note: This will do_something1() for the first set before handling the second set, and so forth. The description you give implies that this should be the case, but it's explicit here.