Profile / benchmark wordpress on apache?

It is impossible to just guess what the performance issue is, it might not be related to a web server at all.

Your options with WordPress are (scope ascending):

  1. PHP runtime profiler / debug code for basic timing of WP load.
  2. PHP engine profiler (binary extension), such as Blackfire.
  3. Full stack profiler (server software, etc), such as New Relic.

You should start with PHP runtime to figure out if your time sinks within WordPress page load. It might or it might not. From there it depends if you need to be further troubleshooting PHP or the problem actually is with web server running it.