It appears that @arr are the nodes in a graph, and %hash are the weights of all the edges in the graph. (It's undirected, and fully connected.) So now you want to sort the nodes — but by what, exactly, is not clear. With the information given, I would surmise that the values for each node come from the weights of all its edges. If so, then the following:

use strict; use warnings;
my @arr = ('foo','bar','qux','foo1','bar1','qux1');
my %hash =
(
    '4-5' => '0.750',
    '0-4' => '0.167',
    '0-2' => '0.600',
    '2-3' => '0.200',
    '2-4' => '0.300',
    '0-3' => '0.300',
    '2-5' => '0.400',
    '1-2' => '0.550',
    '1-5' => '0.273',
    '3-4' => '0.300',
    '1-4' => '0.182',
    '0-1' => '0.917',
    '3-5' => '0.400',
    '0-5' => '0.250',
    '1-3' => '0.300'
);

my @nodew;
for ( keys %hash )
{
    my $v = $hash{$_};
    $nodew[$_] += $v for /(\d+)/g;
}

my @sorted_node_indices = sort {
    $nodew[$a] <=> $nodew[$b]
} 0 .. $#nodew;

my @arr_sorted = @arr[ @sorted_node_indices ];
[download]

Update: I see, you want, for each node, the list of its neighbor nodes sorted by edge weight.

use strict; use warnings;
my @arr = ('foo','bar','qux','foo1','bar1','qux1');
my %hash =
(
    '4-5' => '0.750',
    '0-4' => '0.167',
    '0-2' => '0.600',
    '2-3' => '0.200',
    '2-4' => '0.300',
    '0-3' => '0.300',
    '2-5' => '0.400',
    '1-2' => '0.550',
    '1-5' => '0.273',
    '3-4' => '0.300',
    '1-4' => '0.182',
    '0-1' => '0.917',
    '3-5' => '0.400',
    '0-5' => '0.250',
    '1-3' => '0.300'
);

my %edgew;

for ( keys %hash )
{
    my( $from, $to ) = @arr[ /(\d+)/g ];
    $edgew{$from}{$to} = 
    $edgew{$to}{$from} = $hash{$_};
}

for my $from ( sort keys %edgew )
{
    print "$from:\n";
    print "\t$_ : $edgew{$from}{$_}\n" for 
        sort { $edgew{$from}{$a} <=> $edgew{$from}{$b} }
        keys %{ $edgew{$from} };
}
[download]