Oh so you don't actually have access to the files themselves. That's unfortunate. Then I fear you're at the mercy of whoever wrote that browser plugin.

Subtitles are glorified text files, each line has a timestamp, like

00:07:34,480 --> 00:07:38,080 Exterminate! Exterminate!

This tells the player to display the line from 7 minutes, 34 seconds, 480 milliseconds until 7 minutes, 38 seconds, 80 milliseconds.

If you want the subtitles to be earlier, you're subtracting from those times so it's minus, if you want them to be later, you're adding so it's plus.

I don't know what software you are using and if they are indicating the direction of error or correction, but I trust if you try it you'll figure it out.

Put on media with properly synchronized subtitles. Change to plus five. Are the subs behind or ahead of audio? Make a note. Now you know.

So you just want a better system than "completely at random". Fair enough. Here's what I'd do:

1. After the first line is uttered, see if the subtitles are late or early. Pause.
2. Adjust the delay, let's say -1.0 second (or more if the difference between them felt really long).
3. Skip back to before the line was heard, and play. Is the gap better or worse? If it's worse, then you need to change the delay in the other direction (+1.0 instead of -1.0). If it's better, the direction you used was good.
4. Repeat step 3 until you get that one line synced perfectly.
5. Enjoy the rest of the movie/episode.

Disclaimer: I've never had this problem (that I remember), subtitles have always been either synced or completely out of my control. I don't use 'alternative means' much.

Every video player has a slightly different codebase. I imagine all of these are going to be a hair different when it comes to timing.