Hi Maya,
It looks like you cannot symbolize (change style) based on a date field in ArcGIS Online. You may be able to create the symbology on the desktop end when you publish it. And you may want to make a string field (or numeric field) that would reflect the records in your date field.
Best bet is to make three different numeric fields (month, day, and year) and then you can change the style in ArcGIS Online freely of these three fields.
There are really a lot of combinations you can do in order to get around this hurdle. It will all depend on your application and what you need to do.