0.00043 times 0.003 and then express it as 4 significant figure
I always used to teach that there were two kinds of zeros: significant zeros and necessary zeros.
A significant zero really means that the value in that place really is zero; e.g. in the number 1207, the zero is significant because it really does mean there are zero 10's.
And the rule is that you can't call a number a significant zero until after you have passed (moving from left to right) the first non-zero number. This stops you, for instance, from writing 1207 as - say - 001207 and counting the first two zeros as significant ones. And it also means that in a number like 0.00203, there aren't four significant zeros; there is only one.
A necessary zero is there to make the number the correct size, and doesn't always mean that there really is a zero in that place. So if we say, for example, that the population of a city is 249 000, the last three digits are necessary to make the number the right size - without them it would simply be 249 - but they don't mean that there really are no hundreds, no tens and no units. It probably means that you've simply been told the population to the nearest thousand people.
So in the example above, 0.00203, the first two zeros after the decimal point are necessary zeros, but they're not significant for the reason I've given above - you haven't yet passed a non-zero number. (Incidentally, the zero before the decimal point is neither necessary nor significant - we only write it to make the decimal point a bit more visible.)
In the answer to the OP's original question, 0.000 001 290, the final zero isn't necessary to make the number the right size, and the only reason for including it, therefore, is that it really does mean zero - hence it's significant.
Of course, there are still problems. What if the population of a city is exactly 249 000 at a particular moment in time? If it is, then these three zeros are significant, and there are therefore 6 sig. figs. in this number, not just 3. How can you tell? The answer is that you can't, without further information. You've just got to take what seems the most sensible line.