print join "\n",
      map { $t = $_ ;
            join("\t", $_."\t",
                map { $data{$_,$t} ? "X" : " " } @days
            )
          } @time;
[download]

This code differs from yours in several regards:

• no redundant use of topicalizing single-scalar for to print
• the time as first tab-separated item of the inner join
• for each time an own line

To me this looks like misusing map. Better use it solely to calculate results of values from a list by one or very few statements and no nested map, and putting the results into a new list or array at the same time.