The phrase "in the first iteration of your while loop" made me think of something other than the way I would do it, but the idea is at least very similar. Like this:

$sth->execute();
my @ary= $sth->fetchrow_array();
if(  @ary  ) {
    open(MAIL,"|$mailprog -t");
    print MAIL ...
    do {
        print MAIL join ("\t", @ary), "\n";
    } while(  @ary = $sth->fetchrow_array()  );
    print MAIL "\nEND OF SCHEDULED EVENTS!\n";
    close(MAIL);
}
$sth->finish();
[download]

I'm going to go out on a limb and assume akm2 doesn't have a million appointments a day, in which case we may as well use selectall_arrayref:

my $results = $dbh->selectall_arrayref($sql_statement);
if (@$results) {
 open...
 print...
 print ... join("\t", @$_),"\n" for @$results;
 print...
 close...
}
[download]