Sublime Merge for Linux doesn't pop up version upgrade messages for this very reason (that you need to use the package manager on your system to do it for you). I would imagine that Sublime Text will go a similar way before too long. I'm actually surprised that it still does (perhaps worthy of a mention to @wbond as it may have been overlooked?)
The downside to this in the Merge case is that if you happen to be using the tarball version, you need to perform your own checks to see if it's updated or you end up running an older version because you don't know it's changed and there's no mechanism anywhere to tell you.
That said, as @kingkeith mentions, at the moment the Download button should open a browser and take you to the download page, so if that's also not happening, there is something outside of Sublime amiss there, I would think.