How do I calculate the number of hydrogen atoms in $\pu{0.5 mol}$ of hydrogen gas? I presume the answer would probably somehow use the equation $$n = \frac{m}{M},$$ where $n$ is the amount of substance, $m$ is the mass, and $M$ is the molar mass. However, I'm not sure how to calculate it.
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There is no need to use $$n = \frac{m}{M},$$ since you already have been given the amount of substance. You would need to use the equation, if you were given the mass $m$, e.g. as $\pu{0.5 mg}$.
If you want to calculate the number of hydrogen atoms in $\pu{0.5 mol}$ hydrogen gas, then you should consider that they are diatomic molecules, $\ce{H2}$. The unit mole is represented by the Avogadro constant and is $$N_\mathrm{A} = \pu{6.02214076×10^23 mol−1} \approx \pu{6.022×10^23 mol−1}$$ Each hydrogen molecule has two hydrogen atoms, therefore \begin{align} n(\ce{H}) &= 2 \cdot n(\ce{H2}),\\ N(\ce{H}) &= n(\ce{H}) \cdot N_\mathrm{A},\\ N(\ce{H}) &= 2 \cdot n(\ce{H2}) \cdot N_\mathrm{A},\\ N(\ce{H}) &= 2 \times \pu{0.5 mol} \times \pu{6.022×10^23 mol−1},\\ N(\ce{H}) &= \pu{6.022×10^23}. \end{align}
