Remotely I ping my connection from work. I have a script that checks and alerts me when the pings start failing. I use a domain name and a dynamic DNS service (zoneedit.com) to make it easier to connect from work without needing to know the IP address.
Also, I use
Paessler PRTG which monitors and logs the throughput of the router with nice graphs and you can get it to write the logs to a directory which could be a folder of a web server, so you can then view the graphs from anywhere. Doesn't show a definite downtime, but you could set off some kind of download that will run 24/7 and where it drops may be an indication of failure.
However, if you've got upstream problems, you want to keep an eye on the upstream power levels of the modem. Chances are it's near the limit in which case you just need to pester them to come out and make it so it's not near the limit. Make sure they send a real tech and not a monkey though

. Probably better off with one of the support guys here than calling support. The problem I find is they send someone out and tweak one thing but doesn't fix the real problem, and then send another who tweaks something else and still doesn't fix the problem.
Upstream problems appear to be where the line quality from the modem to the "other end" is of poor quality. Upstream works by the modem generating the signal and if the line is poor the modem has to work harder to get the signal through. Think of it like shouting to make yourself heard is a noisy room. Make the noise go away and you fix the upstream. The fix is usually outside, either in the cable or the green box down the road. NTL seem to assume the problem is in the house. The problem is getting this message through to NTL (took me a year to do that. I had the same symptoms with the modem rebooting all the time, often down for days and weeks at a time).