Could be several things. I haven't really seen issues with Virtuozzo based VPS' either OpenVZ or normal Virtuozzo running Apache and Cpanel as long as the box has at least 512MB RAM.
(Somewhat contradicting) On the other hand, I agree that it is probably running out of memory. You need to tune apache for the correct amount of load/memory usage. Do NOT use the Apache memory limit in Cpanel on a VPS. Next suggestion would be to tune MYSQL correctly as I am sure it is probably idling at 70MB+ memory.
Next step would be to turn off all of the additional crap that CPanel turns on.
Honestly, over the last couple of years and the amount of VPS' and problems we have seen with memory issues...I would never recommend CPanel for a VPS. It is a memory hog and very unforgiving when you have only the ability to burst with limited swap.
My final recommendation if tuning and removing additional services doesn't work. (It should in most cases) Would be to reformat the VPS and get a DirectAdmin license
I know following this post, I am going to get several, "Well my VPS runs fine with CPanel". Be grateful! In most cases it doesn't without alot of tuning and a very forgiving VZ configuration to allow extended bursts. On Xen, you are even worse off. There isn't a burst with Xen and most providers don't provide swap with Xen, so if you hit that 512MB....box goes down for a reboot.
Simplest place is to do a search on WHT for: apache tuning or httpd.conf tuning. There are several howtos that are very useful in the VPS section, many provide exactly what you need.
----------------------------------------------------------
I would also recommend checking the logs as Decker mentioned as well. It could also be that Apache is core dumping and restarting. (Possibly due to script issues, memory problems, too many threads.....etc/etc
Good luck.