Which orbital would the electron of a ground state hydrogen atom occupy?

The electron would occupy the $\text{1s}$ sublevel in the $\text{1s}$ orbital.
Every hydrogen atom has the atomic number $1$, which is the number of protons in its nucleus. If the atom is neutral, it will also have $1$ electron.
In the ground state, the only place that electron can go is in the $\text{1s}$ sublevel. There is only one orbital in the $\text{1s}$ sublevel, which is an $\text{s orbital}$. So its ground state electron configuration is $\text{1s"^1}$.