On your local system, create a skeleton of what you want. For example, if you want to copy file foo to remote location /etc/foo, then you need to create an etc directory and then put foo into it. Then tar the skeleton. Now you can do this via cron as suggested by @Anthon in the comments to the question above.
Step by step:
On the remote host, create script like this:
#!/bin/sh
DROP="/home/YOURUSERNAME/drop.tgz"
if [ ! -s /home/YOURUSERNAME/drop.tgz"$DROP" ]; then exit 0; fi
cd /
tar -pzxf /home/YOURUSERNAME/drop.tgz"$DROP" && rm "$DROP"
On the remote host, add that script to a cron job that runs as root.
On your local host, create the skeleton and populate the files you want copied:
mkdir -p etc
mkdir -p var/www
cp -a foo etc/
cp -a bar var/www/
tar -pzcf drop.tgz etc var
scp drop.tgz REMOTEHOST:
rm -rf drop.tgz etc var
The drop.tgz will be extracted when the root's cron next runs. Note, this will overwrite all kinds of things that you might not like.
A safer option, assuming there are only a few files you need to modify, would be to make your user account have write access to them (chown or both chgrp and chmod g+w), then you can scp them directly.