OK. You're making real progress.

You can simplify your life by moving the sudo into your script.
The cat is not needed and slows down your script (although not by a lot in this case); it just gives you an extra (unneeded) process and causes your script to read the entire contents of your input file twice (when only once is needed) and to write the entire contents of your input file (when it is not needed at all).

Assuming that the differences between the date stamps is only used for debugging (and isn't needed for further processing), then we can get rid of the invocations of date inside the loop (as you suggested) so we can just perform string comparisons. To do that we need to convert the month name into a month number and we need to use the awk substr() function a few times to rearrange the text in field 4 into the format:
YYYYMMDDHH:MM:SS. We could have removed the colons from this, but it would take two additional calls to substr(), so it was simpler to add the colons to the date output when setting RefDate. So if you rename your script to something like monitor-rewrites and invoke it with something like:

Code:

./monitor-rewrites /var/log/httpd/rewrite.log "10 hours"

where monitor-rewrites contains:

Code:

#!/bin/ksh
IAm=${0##*/}
if [ $# -ne 2 ]
then	printf "Usage: %s Logfile TimeRange\n" "$IAm" >&2
	printf '\twhere "Logfile" is a pathname to the logfile to be\n' >&2
	printf '\tprocessed and "TimeRange" is something like "1 day" or\n' >&2
	printf '\t"4 hours" spcifying which records from "Logfile" are\n' >&2
	printf '\tto be processed relative to the current time.\n' >&2
	exit 1
fi
Logfile="$1"
TimeRange="$2"

sudo awk -v RefDate=$(date -d "new - $TimeRange" '+%Y%m%d%H:%M:%S') '
BEGIN {	print "BEGIN"
	b2m["Jan"] = "01"; b2m["Feb"] = "02"; b2m["Mar"] = "03"
	b2m["Apr"] = "04"; b2m["May"] = "05"; b2m["Jun"] = "06"
	b2m["Jul"] = "07"; b2m["Aug"] = "08"; b2m["Sep"] = "09"
	b2m["Oct"] = "10"; b2m["Nov"] = "11"; b2m["Dec"] = "12"
}
{	# Sample field 4 data:
	#	[26/Jun/2014:13:41:16
	#	[dd/abm/YYYY:HH:MM:SS
	#	000000000111111111122
	#	123456789012345678901
	# Convert to:
	#	YYYYmmddHH:MM:SS (The colons are left in to reduce the number of
	#	substr() calls needed to reformat the date and time.
	LogDate=substr($4, 9, 4) b2m[substr($4, 5, 3)] substr($4, 2, 2) \
		substr($4, 14, 8)

	# Uncomment the following for testing
	# print "R:" RefDate " L:" LogDate " Line: " $0

	if(RefDate < LogDate) {
		if (/rewrite +.?(assets\/.+.? +-> +.?sites\/default\/files\/|sites\/default\/files\/.+.? +-> +.?assets\/)./) {
			print $4, $5, $10, $11, $12, $13
		}
	}
}
END {	print "END"
}' "$Logfile"

it should run a little bit faster since your awk script doesn't have to start up a shell to run date for every line your process in your file.

As I said before, I can't fully test this since the date utility on my system doesn't support the -d option and nothing in the sample input you have provided matches the ERE you're using to select lines to print, but this should be close to what you want.

I hope this helps.