It changes the style synchronously, and you will notice that if you read back the value on the next line and show it.
$("#test").css("display", "inline");
alert("Showed element!" + $("#test").css("display"));
But the change of the style object triggers a repaint request message to the page renderer, and that is handled as soon as the browser becomes idle, which is after this script routine has ended.
It depends on the browser, though. In Edge it works fine, and the element is shown right away but in Chrome and Vivaldi it is not.
Another test to see how browsers handle this:
If you resize the browser window, JSFiddle will scale (each of the areas keeps the same relative size). If you resize the Vivaldi browser with the alert open, it doesn’t do that. In fact if you make it small, then show the alert, then make it larger, you just get a grey area in the new space until you close the message box. In Edge, the fiddle will just resize in the background, even though the entire browser window is grayed out, so it’s not just a matter of the time of processing, but more that Chrome completely freezes the page when an alert is open.