Here is one way (using today's last month but you could set it otherwise):

use Date::Calc qw( Today Days_in_Month Day_of_Week Day_of_Week_to_Text
                   Add_Delta_YM );

my ( $year, $month, $date ) = Today();

( $year, $month, $date ) = Add_Delta_YM( $year, $month, $date,
                                         0, -1 );

my $day = ( Days_in_Month($year,$month) )[-1];

print $day, "\n";

print Day_of_Week_to_Text(
                          Day_of_Week($year,$month,$day)
                          ), "\n";
[download]

Additions above (I realized you might mean day of week and not date)!

Update 2: D'oh! Re-corrected version to get last month's is now above.