Server hitting high loads and becoming very slow
Solution 1:
256 MB of RAM and Apache is the hog... I bet that you're causing a lot of swap activity as requests are handled by different instances of the daemon. I recommend using atop for an interactive view of memory, cpu, and disk / swap activity.
It's possible that MySQL doesn't have enough memory to work with as well -- if MySQL is configured to use a lot of memory but doesn't have it available then it may be using the oldest memory sections first which are swapped to disk.