The equivalent on AWS Cloudfront would be breaking the cache, so all users would see the most recent version.
I'm not sure if this would work on IPFS because it seems more like a P2P network and there might be many different (and old) versions of the site at different locations in the system, with no way of controlling who sees what.
IPFS is p2p, but all the content is accessed only by a hash, so the same address always refers to the same version; when you update the site, you're actually creating a new address, leaving the old untouched.
Using Cloudflare, you can tie a regular domain to one of these addresses, but that "tie" is only stored there, so when you push a new version, you can update the Cloudflare settings, and everyone connecting to the domain will see the new version "immediately", even if the old version still exists in the p2p
There is a number of services, the price seems to be around $0.3/(gb*month)
If you just use IPFS directly with a non-CDN'd, non-cached http gateway, the latency will probably be prohibitive for regular web stuff.
Edit: found it. https://www.reddit.com/r/ipfs/comments/al8emd/cloudflare_ban...
This means that even if your content in ipfs is popular it won’t be fast.
If you want to really ensure it's live you should run at least one node that pins it yourself (or pay someone to do that)