How can Hydrogen NMR be used to distinguish between these two isomers?
In the absence of any other configuration anchoring factors, I would not expect the conformation of the two structures to be reasonably stable at room temperature to be able to distinguish both by NMR. You may well need lower temperatures to freeze chair inversion and create a set of distinct, non-averaged peaks.

This issue aside, it is very easy to distinguish the two once you know you have a stable conformation. The key is the Karplus relationship which allows you to link a vicinal coupling constant to the dihedral angle between the two hydrogens. In general, a $^3J_\ce{HH}$ coupling constant will be very high ($^3J_\ce{HH} &gt; \pu{9Hz}$) if the dihedral angle is either $0^\circ$ or $180^\circ$ — typically, if the hydrogens are anti-configured. On the other hand, the coupling constant becomes minimal for values around $90^\circ$ ($^3J_\ce{HH} &lt; \pu{2Hz}$). For a $60^\circ$ gauche arrangement, a coupling constant of $^3J_\ce{HH} \approx \pu{3 Hz}$ is expected.

So the coupling between $\ce{H{1}}$ and $\ce{H{2}}$ would be around $\pu{9Hz}$ or $\pu{10Hz}$ in the first case but only around $\pu{3Hz}$ in the second — if the configuration is long-living on the NMR timescale.