Pretty simple.

use strict; use warnings; 
my @weekdays = qw(
Today
Yesterday
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
Saturday
Sunday
);

my %lookup; 

  my @backs; 
  use Time::Local;

  my ($sec,$min,$hour,$mday,$mon,$year,$wday,$yday,$isdst) =localtime(
+time) ; 
  # move to noon so DST  changes dont kill ya 
  my $now_fix12=timelocal( 0, 0, 12, $mday, $mon, $year);

  for my $back (0..6) { 
    my ($sec,$min,$hour,$mday,$mon,$year,$wday,$yday,$isdst) =localtim
+e(time-24*60*60*$back) ; 
    push @backs,[$wday,$mday]; 
    $lookup{$weekdays[$wday+2]}=$mday; 
    }

  $lookup{'Today'}=$backs[0][1]; 
  $lookup{'Yesterday'}=$backs[1][1]; 

  for my $wday (@weekdays) { print $wday.' '.$lookup{$wday}."\n"; }
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Edit:code fixed to fit changed OP

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use Time::Piece;
use Time::Seconds;

sub show {
    my ($date, $label) = @_;
    $label ||= $date->fullday;
    say join ' ', $label, $date->month, $date->mday;
}

my $today = localtime;

show($today, 'Today');

my $yesterday = $today - ONE_DAY;
show($yesterday, 'Yesterday');

for my $minus (0 .. 7) {
    my $prev = $today - $minus * ONE_DAY;
    show($prev);
}
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($q=q:Sq=~/;[c](.)(.)/;chr(-||-|5+lengthSq)`"S|oS2"`map{chr |+ord
}map{substrSq`S_+|`|}3E|-|`7**2-3:)=~y+S|`+$1,++print+eval$q,q,a,
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G'day james28909,

You can do this sort of thing fairly easily with Time::Piece and Time::Seconds.

There were some points I wasn't too sure about from your spec:

• Did you want the named days to be presented sequentially, or work from just the last week, which would mostly present two sequences. In the code below, I've assumed the latter.
• Should Today and Yesterday be duplicated in @weekdays or are they separate dates. I've assumed the former.
• Was the second Sunday an intended duplicate. I've assumed not. If you do want a second Sunday, probably subtracting Time::Seconds' ONE_WEEK constant from the first Sunday will do what you want.

Here's code, showing the basic techniques. Bear in mind my stated assumptions. Note that the BEGIN block is just to get a point of reference before Time::Piece reimplements localtime. Also, in case the dates and times look a bit screwy, my timezone is currently UTC+10:00.

#!/usr/bin/env perl -l

use strict;
use warnings;

BEGIN { print scalar localtime }

use Time::Piece;
use Time::Seconds;

my $t = localtime;
my @weekdays = ($t->mday);
my %lastweek = ($t->day => $t->mday);

$t -= ONE_DAY;
push @weekdays, $t->mday;
$lastweek{$t->day} = $t->mday;

for (1 .. 5) {
    $t -= ONE_DAY;
    $lastweek{$t->day} = $t->mday;
}

push @weekdays, @lastweek{qw{Sun Mon Tue Wed Thu Fri Sat}};

print "@weekdays";
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Output:

Sat Apr 29 16:20:18 2017
29 28 23 24 25 26 27 28 29
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Had you also wanted the second Sunday, the output would have been:

29 28 23 24 25 26 27 28 29 16
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If you had wanted the eight days, with two Sundays, to be sequential, the output would have been:

29 28 16 17 18 19 20 21 22 23
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I'm guessing you can probably work it out from here; however, if you need further help, please provide expected output.