It is possible to program many devices with static IP.. I do not know about your modem, but mine works that way ...
Now, There are two ways to do what you want.. In some cases it is indeed possible to connect to a modem and reset it, but if it's not talking to the internet,,, That's not going to work
If you have a time of day when you can always get by without it (Say 2 am or 4 am) you can put a 24 hour timer, Shut it down for 15 mintues... (analog timers) or 1 minute (digital timers) and when it powers back up it's reset. Of course you must make sure this happens AFTER the nightly ISP screw up.
Another thing... A 3-5 Minute timer on power frail, keep it off for several minutes incase the power fail also affects the ISP side of things.