As the measurement postulate says, if you projectively measure a qubit, initially in a state $|\psi\rangle$, in the basis $\{|+\rangle,|-\rangle\}$, you will get the state $|+\rangle$ with probability $|\langle+|\psi\rangle|^2$, and similarly for $|-\rangle$.
For the particular implementation you mention, a two-level atom whose eigenstates are the logical $|0\rangle,|1\rangle$ states, there is no general, useful, real physical quantity${}^1$ represented by the operator $$X=|0\rangle\langle1|+|1\rangle\langle0|$$ whose eigenstates are $|+\rangle$ and $|-\rangle$ (check it!). To do a projective measurement on that basis, the standard (though not necessarily unique) procedure is to apply a $\pi/2$ Rabi pulse which will bring $|+\rangle$ to $|0\rangle$ and $|-\rangle$ to $|1\rangle$, and measure in the computational basis. One can then apply an inverse pulse if needed.
There are other implementations, however, where this basis has a more physical significance. For example, if your logical states are the up and down states of a spin-$\frac{1}{2}$ particle measured along the $z$ direction, then $X$ is the spin along the $x$ direction (which is no coincidence).
${}^1$ For any given atom, though, you can probably find detectable physical properties of interest. If, say, $|0\rangle$ is an $s$ state and $|1\rangle$ is a $p_z$ state, which may very well be the case, you'll find that the $|\pm\rangle$ states are localized towards either pole. A measurement of position above/below the $xy$ plane will closely approximate an $X$ measurement in most such circumstances. Similarly, a measurement of momentum going to positive or negative $z$ will approximate a measurement along $Y=i|0\rangle\langle1|-i|1\rangle\langle0|$, whose eigenstates $|\pm i\rangle=\frac{1}{\sqrt{2}}(|0\rangle\pm| i\rangle)$ look like $e^{\pm ikz}$ near the origin.
