I don't have exact figures, but each site probably gets a few hundred hits/day, and the mail server probably gets hit every few seconds. The OP made a sweeping "not an option" statement though, not "on a popular site hosted on a VPS".
Let me re-phrase. Using Apache + mod_php in a RAM constrained environment is not an option on any site that receives more than a dozen requests per second, assuming proper configuration. However, that figure is unbelievably limiting. The site can't be used for anything other than a very low traffic internal site or a dead blog. Even a tiny spike in traffic will destroy your setup. If you switch to Apache + PHP-FPM you can increase your capacity 10X with the same hardware. Literally. Most resources on a website are static and loading PHP for all of them is a pointless waste of RAM.