Hey there! I'm sorry to hear about these issues you're experiencing with the VPS server. One of the possibilities for this behavior would be a Denial of Service attack against Apache on the system, as that sounds a lot like what you are describing. If Apache is not able to function properly, but works for a brief time after a restart, that often indicates a DoS is happening.
We have a guide here that will help you recognize if this is what is happening, and offer some solutions:
Question How can I tell if Apache is experiencing a DDoS (Distributed Denial-of-Service) attack? Answer If Apache is experiencing a DDoS attack, you may notice that your server's Web sites are ti...
support.cpanel.net
so I'd recommend checking that out. If that isn't the issue, just let me know.