Read the file in paragraph mode as others have suggested then process each paragraph by spliting into words and the separating whitespace (by capturing the whitespace you split on), doing the substitution when appropriate.

use strict;
use warnings;

open my $inFH, q{<}, \ <<EOD or die qq{open: <<HEREDOC: $!\n};
Class control
priority 5 
Class voip
priority 30 
Class video
priority 40 


Class control
priority 10 
Class voip
priority 25 
Class video
priority 45 
EOD

{
    local $/ = q{};
    while( <$inFH> )
    {
        my $foundPriority = 0;
        $_ =
           join q{},
           map {
                   unless( m{\bpriority\b} )
                   {
                       ;
                   }
                   elsif( $foundPriority ++ )
                   {
                       s{priority}{bandwidth};
                   }
                   else
                   {
                       ;
                   }
                   $_;
               }
           split m{(\s+)};
        print;
    }
}

close $inFH or die qq{close: <<HEREDOC: $!\n};
[download]

The output

Class control
priority 5 
Class voip
bandwidth 30 
Class video
bandwidth 40 

Class control
priority 10 
Class voip
bandwidth 25 
Class video
bandwidth 45
[download]

I hope this is useful.