Have you asked the peoples that are hosting your site if they have a solution?
Most hosting companies have solutions for this (even on Windows servers)
If the hosting company has no solution for this (what I seriously doubt) you can call the script externally from another server that has cron possibilities or have it called by the first visitor on that day.
I any case you will want to include a check in your script that your script only runs on the set date (you already have that) and also make sure that calling the script more than once on that day is no problem.
PS : if the script is called by the first visitor you will have to consider the fact that you could not have any visitor that particular day, so the script will have to run whenever the next visitor comes along.