You are looking for Symbolic references. The following code does what you intend; However, I would think a dispatch table would be a cleaner, safer and less bug-prone solution to your issue. I, of course, have no idea what that issue is. Please read Why it's stupid to use a variable as a variable name before you use the code.

#!/usr/bin/perl
use strict;
use warnings;

my $sub = 'do_this';

{
    no strict "refs";
    print $sub->("Fred");
}


sub do_this {
   my $var = shift;
   return "My name is $var";
}

sub do_that {
   my $var = shift;
   return "My name is not $var";
}
__END__
My name is Fred
[download]

And with a dispatch table, which leaves strict intact:

#!/usr/bin/perl
use strict;
use warnings;

my %dispatch_table = (
    do_this => sub {
            my $var = shift;
            return "My name is $var";
        },
    do_that => sub {
            my $var = shift;
            return "My name is not $var";
        },
    );

my $sub_name = 'do_this';
print $dispatch_table{$sub_name}("Fred");
__END__
My name is Fred
[download]

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