I assume you mean inclusive of 1000 and 2000.
Let's put the number 1000, which has 3 zeros, aside 
for a moment and look only at the 1000 whole numbers
from 1001 to 2000. 

There are 1000 ones digits.
One tenth of them are 1's, one-tenth of them are 2's,
one-tenth of them 3's, etc., one-tenth of them are 9's.

One-tenth of 1000 is 100.
So 100 of the ones digits are 0's.  Similarly
100 of the tens digits are 0's, and
100 of the hundreds digits are 0's, so there are
300 zeros from 1001 through 2000.  There are no
zeros among the thousades digits because they are
all 1's but the last one, which is a 2.

So that makes 300 zeros, plus the 3 zeros from 1000.

Answer 303.

Edwin