Can I always have a chance to use Stupefy without opponent response right after I use Hextech Ray in a Showdown?

No, you cannot always do this without your opponent getting a chance to respond — but it depends on whether you retain priority.

Here's the breakdown:

Stupefy has [Reaction] (not Action), so it can be played on the same Chain as Hextech Ray — that part is fine. The issue is priority. Under the tournament shortcut (Rule 503.9.b), when you put Hextech Ray on the Chain you are assumed to have passed priority unless you explicitly announce you're retaining it. That means your opponent gets the first window to act after Hextech Ray, before you can play Stupefy.

If you explicitly announce you're retaining priority after playing Hextech Ray, you can immediately play Stupefy (as a Reaction on the same Chain) before giving your opponent a turn. After Stupefy resolves, the Chain still has Hextech Ray on it (since it resolves LIFO), so your opponent can then respond — but Stupefy's effect (giving -1 Might and drawing) will already have happened.

Bottom line: By default your opponent gets a window. To play Stupefy without them acting first, you must verbally retain priority after Hextech Ray.