When you just want to "push" your local website without getting offline long, you can use a temporary dir.
TODAY=$(date +%y%m%d)
NEWCODE=/var/www/vhosts/anotherwebsite/docs_${TODAY}
OLDCODE=/var/www/vhosts/anotherwebsite/docs_old
rm -rf ${NEWCODE}
cp -rpf /var/www/vhosts/website/httpdocs ${NEWCODE} || exit 1
# some checks ?
cd /var/www/vhosts/anotherwebsite/ || exit 1
mv httpdocs ${OLDCODE} || exit 1
mv ${NEWCODE} httpdocs
Between the moves you will be unavailable. When that is a problem, you might want to make a work_in_progress.html file, rename that file to httpdocs/index.html, remove all other files and copy the new files after this (the correct index.html file as the last one). But this seems to fancy, stick with the short hickup in the solution above.