How to calculate atoms of *each* element in a compound
Think of the mole as a SI prefix, like 'kilo' or 'mega'. If you have 5k $\ce{H2SO4}$ molecules, you will have 10k hydrogen atoms, not 5k. Just as 'kilo' means '1000', a mole means $6.02 \cdot 10^{23}$ of something.

Therefore, the first approach gives the correct answer, but I don't understand what you mean by 1gr/mol. Grams are out of the picture here.