OK, let's take the sine wave completely out of the problem, it's actually only obscuring the picture.
What we're looking at is placing a precise event in time. Since we started in 'theoretical land', let's keep it that way.
The 'event' to look for is a mathematical impulse. This is an infinitely thin spike, or pulse, that has a finite amount of energy. The energy is evenly spread across the whole spectrum (frequency). Real impulses have finite width, due to being bandwidth limited (by whatever physical situation they're in).
Let me first give a real world example (albeit electromagnetic - including radio frequencies): when you flick on a light switch, or otherwise create a spark, it is an impulse of current flow, which radiates out electro-magnetic waves over a broad band f frequencies. You radio (particularly AM) will pick this up pretty much whatever channel you're on and give you a click. Even if it's battery operated (no AC interference). This (somewhat) demonstates a real word impulse and it's broad band frequency content.
The thing about an impulse is that in time it is symmetrical so it is easier to see where it's center is - i.e. see where the actual event time is. To do this for the burst sine start, you have to remove the original signal to find the 'ringing', then find the center of that. The problem is you don't know where or how the sine starts - as that's what you're trying to find out.
Now I could do some more CoolEdit diagrams etc, but again, sampling has nothing to do with it. What you would do really is, pass this ideal impulse though what ever filter you like and see if you can locate the position of the original impulse.
I think what you'll find is that you can. Which I think ultimately answers the question - hopefully finally. I haven't looked into this much with real (analog) signals with real filters though, but the same argument should work.
If you can locate an impulse, you can locate the start of your 'ideal' burst.
