Please, use <code> </code> tags when posting your code.

As to the question, a quick fix solution you could try is getting the data from the database sorted descending by time, so that the newest times come first. Then, you can push the first occurence of a job name onto a list of data rows, and when the while loop ends, you can print the rows.

# I'm not sure if it's the correct statement
# for getting the data sorted newest first, test it first
&DBEXEC("select proc_eventvu.job_name, jobst.status, proc_eventvu.stam
+p, proc_eventvu.text
        from proc_eventvu, jobst
        where proc_eventvu.joid = jobst.joid
            and proc_eventvu.event in (112,$eventnumber)
            and jobst.status = $statusnumber
        order by proc_eventvu.stamp desc");

my %already_processed;
my @results_lol;

while(@status = &dbnextrow($dbproc)) {
    $name = ($status[0]);
    $jstatus = &JOBSTATUS($status1);
    $estamp = ($status2);
    $etext = ($status3);
    
    # only the job data for a jobname
    # that wasn't processed before 
    # make it to the result list
    unless (defined $already_processed{$name}) {
        push @results_lol, [($name, $jstatus, $estamp, $etext)];
        $already_processed{$name} = 1;
    }
}

# here, if you want the rows sorted by name, you could 
# write some necessary code

for my $row_ref (@results_lol) {
    my ($name, $jstatus, $estamp, $etext) = @$row_ref;
    write;
}
[download]

It's untested, but try if it works. And I wholeheartedly second the suggestion about  use strict; use warnings;, as well as the advice about the JOBSTATUS subroutine.

Regards,
Luke