The long TTFB is coming from your CMS doing it’s back-end processing (usually database but could be other external back-end calls or server configuration). It looks like you are running on top of prestashop. I’m pretty sure we’ve had at least one other user here with similar issues (I’ll poke around search a bit).
You should be able to use a system file cache - most shared accounts have permissions to write to the local filesystem. Memcached requires an install and is probably outside of your permissions though.