Change date on icon click event
There are two ways to go about this.
Pure JS
You can send the date shown to the JavaScript either via drupal_add_js or as a data attribute and let the JavaScript manage the date changes. You would need to change the current date +/- 1 when clicking on a button and figure out the name of the day and the day number of the day and display that. This should be relative simple to do.
AJAX
You can let the use AJAX and send the selected date to a custom menu callback which returns the markup for the new date.
Of the two solutions I would prefer the pure JS version. The Ajax version is more complex to implement and will generate a bigger load on the webserver. It could be a nice solution if you want to do advanced theming and don't want to place that logic in the JS (like weekends / holidays etc are displayed in different styles)