Is it possible to make Apache load balance hits to a particular directory? If a lot of requests are coming in to just a specific resource, I don't want to cripple the servers ability to respond to other requests.
Load balancing requires multiple servers and some sort of load balancing device that sits in front of the resources being balanced. It doesn't sound like you have the pre-requisites for this scenario. Apache isn't usually the tool used used to balance loads, it is the resource being balanced.
That said, there are ways to configure your system to reduce the problem I guess you face. One would be to rate limit request to a specific resource so that the server would never get slammed with those requests. This would make the API unresponsive some of the time.
Another would be to run a separate server for just that resource, then use apache to proxy or otherwise pass off the load so that it's not busy filling those requests.