What type of Internet do you have, is it dsl? You said you set it up for PPoE? So you have a username and password you have to put into the router to connect? If you don't, it should be set for DHCP, does your router connect to a modem?
The best way to troubleshoot this is, the times your Internet doesn't work on the phone, go to your phones browser and type in the ip for your router (192.168.2.1). If you can get to your router's interface, then you have a problem with your ISP most likely , if you can't get to the interface, then we know it's a local problem.