Apache can certainly handle the requests, but you will want to do some careful planning to ensure that things work as expected.
In order to handle such a high rate of requests with minimal latency, clients MUST be using HTTP keepalives. Keepalives are sometimes disabled in web servers.
You will also want to be very careful on the client-side to not issue a new request if the previous one is still pending. You might also want to disregard a response if it took too long.
If your game requires this request rate, it may be a good idea to limit it to players with low latency to your server (say < 50ms) which will prevent players outside your geographical region (you could always set up new servers for those players elsewhere).
Personally I would always recommend using a FastCGI backend for Apache (or maybe Nginx or some other web server). Then you will need a pool of processes / threads to run your FastCGI. This does not need to run as root (and indeed, should not).
In the longer term though, if the client browsers support it, it might be a good idea to use Websockets.