Your answer is correct. That is the more intuitive approach. If you want to brute force compute it, you can say that you are trying to solve P(draw 00 | see a 0). That is, the probability you draw the 0/0 card given that the side you draw shows a 0.
We can solve this breaking down P(draw 00) into all its conditions:
$ \text{P(draw 00)} = \text{P(draw 00 | see a 0)} \cdot \text{P(see a 0)} + \text{P(draw 00 | see a 1)} \cdot \text{P(see a 1)} $
NOTE that P(draw 00 | see a 1) is 0 so the second term drops out (if you see a 1 you can't have the double zero card).
So we are left with:
$ \text{P(draw 00 | see a 0)} = \frac{\text{P(draw 00)}}{\text{P(see a 0)}} = \frac{\frac{1}{4}}{\frac{5}{8}} = \frac{2}{5} $
The $\frac{1}{4}$ comes from 1 of 4 cards being a double zero, and the $\frac{5}{8}$ comes from 5 of 8 sides having a 0.
You realized the shortcut which is to say that there are only 5 zeroes and 2 of them belong to the double zero card.