Use the dropdown list event onchange to show when the user select any item from the dropdownlist.
The onchange event is a javascript event.
Or we can use the c# SelectedIndexChanged event and you can add the same code you have used inside button click event also in the SelectedIndexChanged event. If your buttion click event is in javascript add the same code inside the javascript onchange event.
Use the following link. It will show how to show the popup on dropdownlist click
Display popup on selecting drop down list item[
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