One solution is to use globals for this situation.

Yes, global (lexical or package) variables will do the trick, but localized package variables would be better:

sub get_time {
    my ($string) = @_;
    local our ($sunrise, $sunset);
    $string =~ /
       sunrise: \s+ (\d+:\d+) (?{ $sunrise = $^N })
       \s+
       sunset:  \s+ (\d+:\d+) (?{ $sunset  = $^N })
    /x;
    return ( \$sunrise, \$sunset );
}
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...or rethink your regexp strategy.

I think this would be the proper course of action here. Using (?{...}) is overkill, and using $^N imposes a requirement for Perl v5.8.0+ needlessly. The OP could use the following simple expression:

sub get_time {
    my ($string) = @_;
    my ($sunrise, $sunset) = $string =~ /
       sunrise: \s+ (\d+:\d+)
       \s+
       sunset:  \s+ (\d+:\d+)
    /x;
    return (\$sunrise, \$sunset);
}
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By the way, why return references to the values? return ($sunrise, $sunset); would make more sense.