First, I have to note, parsing ls is generally not recommended. Its output is meant for human consumption, and this solution will fail for some legitimate file names (like those which contain newline characters, for example)

With that said, the simplest solution does seem to be using ls (as find does not offer a sort-by-modification-date option). For this, you can use pipes and the xargs command to execute the final rename.

In short:

ls -1t | head -n5 | xargs -I {} rename 's/ /_/g' {}

This uses your original command almost exactly, but with xargs to do the final completion. xargs will take in a number of newline-separated arguments, and pass them all to a single rename command.

Caveats

This should work - but parsing ls is fragile. This breaks for file names which contain the newline character; using a command that's meant for parsing like find would be much better.

However, I don't know how to sort by last modification date with find. If someone else knows how to do this, you'd probably end up with something like:

find . -mindepth 1 -maxdepth 1 -print0 | sort <???> -Z | head -z -n5 | xargs -0 rename 's/ /_/g'

If <???> were a valid argument, this command chain would tell find to output files in the current directory separated with a NUL character, and then sort, head, and xargs would all accept this as the deliminator rather than a newline - making the whole process much more consistent. Feel free to comment or edit this if you know how to do this, and want to replace the ls answer with this bottom part.

As already pointed out, parsing ls is fragile. It will break if your file names have any type of weirdness (spaces, newlines, control characters etc). Since you want to replace spaces with _, your suggested approach will fail. A safe way to do this would be:

stat --printf '%n/%Y\0' * | sort -rz -t'/' -k 2 | head -z -n 3 | 
    cut -d'/' -z -f 1 | xargs -0 perl-rename 's/ /_/g'

• stat --printf '%n/%Y\0' * : this will print out the name (%n) of every file or directory in the current directory followed by a slash (/; I am using a slash since that is not allowed in a file name so can safely be used as a separator) and the file's modification time in seconds since the epoch (%Y) and finally a NULL character (\0).
• sort -rz -k 2 : this will take NULL-separated input (-z) and sort it in reverse order (-r) based on the value in the second /-separated (-t'/') field of each line (-k 2).
• head -z -n 5 : keep the first 5 (-n 5) null-separated (-z) lines.
• cut -d'/' -z -f 1 : print the 1st (-f 1) /-separated (-d /) field of null-delimited (-z) data.

I just gave ls -q a try:

IFS='
'
for file in `ls -tr1q|tail -n5`; do
  rename 's/ /_/g' "$file"
done

Does the desired rename, even for filenames with newline or other control characters, although it gives an error:

Unsuccessful stat on filename containing newline at /usr/bin/rename line 59

(with perl v5.20.2)