It's unclear what you mean when you say the linux box 'stops responding'. The linux box is live and running,you can access the internet, but you can't access apache on it from another computer? Is that correct? Something doesn't seem right, that you can access the internet but not apache, and resetting the router gives you access to apache again.
if that's the case, then first thing is open a browser on the linux box and a different windows computer, and surf to the ip address of the linux box, i.e. [198.168.nnn.nnn.
If it doesn't respond on either machine, probably something goofy with apache.
if it works on the linux box and not the windows machine, then apache is running fine but there's something stopping things from the linux box forward. That could be the router, could be your cable, could be networking, could be a firewall.
I'd start with the firewall on the linux box. if it's an off the shelf linux firewall it probably has a gui for the firewall. Figure out how to look at that. Open everything. that's not as insecure as it sounds.
if the firewall is fine, next step is to look at networking. Not sure what could be going wrong, but from the windows box, try to ping the linux box using the ip, and try a traceroute to the linux box i.e. ping 198.168.nnn.nnn (or whatever the windows equivalent is of ping and traceroute). And then on the linux box do a 'ifconfig'. Post the results of those three things. Again, it's really odd that you can get to the internet from the box, but you're suggesting it's basically off the local network.
the other potential culprit is the router. Not sure what would be causing it, but routers can do wonky things.
And another potential culprit is the dns, but not sure that makes sense given what you've said.