Since you're interested in specific city the number of airports you're considering seems limited. You may check if they have some sort of a newsfeed/rss on the airport page. I may be wrong, but airports like to brag about any new routes they managed to attract to the airport so the info should be on their webpage. Check if they don't have a newsletter to sign up for.
Even if the answer is no, still all airports I have ever checked have a page for departures/arrivals schedule (not the next x, but the plan for the whole week). Use a tool to track changes on such pages. As mentioned in a comment to a different answer, there are browser add-ons (I used to use UpdateScanner for Firefox, I don't know if it is still working though) to do that.
Another option you may consider is looking for some newspapers regarding airliners or aviation. If there is anything local enough, they may be listing this kind of information (according to their publishing schedule, so probably once a month).
Either case you'll also get some extra junk but that you can rule out by using filters/rules in your mailbox. Changes tracking on the departures page is probably least vulnerable to false positives (i.e. you're notified when there's actually nothing interesting changed).