You can't just get it from the atomic properties, the electronic properties of a metal are dominated by "solid state"-type considerations, for instance, the fact that electrons live in a band structure rather than something more akin to the usual discrete levels that one learns about in QM 1.
Thankfully, <PERSON> and <PERSON>'s classic book has a long discussion on the work function in chapter 18.
Their formula is $W=-\epsilon_F+W_s$, where $\epsilon_F$ is the Fermi energy, a quantity determined by the density of electrons and the properties of the crystal lattice of the metal; you can work out reasonable approximations to this for alkali metals by using the free electron approximation. $W_s$ is a quantity related to surface effects; for this term <PERSON> and <PERSON> give a model with a dipole moment per unit area of $P$, so that $W_s=-4\pi e P$.
I'm not sure whether you can really get "within a few percent" with such crude techniques, but it's certainly something that's calculable. In particular, getting a good approximation comes down to two things, 1) getting a handle on the band structure of the metal so that you can calculate $\epsilon_F$ accurately 2) having a good model for the surface of the metal.
There's a wrinkle here about how $\epsilon_F$ is defined. 28
One can't just use the usual expression $\epsilon_F=\hbar^2k_F^2/2m$ as one needs to add in a term corresponding to the electrostatic energy of the ions, something like the Madelung constants.
Again, I recommend the book of <PERSON> and <PERSON> for this (and any other just-beyond-basic questions you might have about thinking about electrons in metals and semiconductors).
Just out of curiosity, I dug into the literature a little bit to see what researchers have done.
In 1971, <PERSON> and <PERSON> were able to get work functions of simple metals to about 5% and noble metals to 15%. I think it would be possible to reproduce these calculations today quite easily.
A more recent paper (2005) by <PERSON>, <PERSON>, and <PERSON> uses all-electron first principles calculations, as opposed to the more empirical fitted pseudopotential calculation that <PERSON> and <PERSON> used. My impression from skimming the data is that they only get marginal agreement with experiment; in particular, I noticed that the experimental values seem to have quite a bit of variance in them, consistent with the fact that work function is strongly dependent on surface properties, which are hard to get consistent.
For example, for Aluminum along the 1 1 1 crystal plane, their calculations yield 4.21eV, compared to experimental values of 4.48, 4.24 and 4.33.
By comparison, <PERSON> and <PERSON>'s 1979 calculation gave 4.05 eV and they quoted an experimental value of 4.19 eV.
Apparently the last big review on calculating work functions of metals is this one by <PERSON> et al from 1979. I didn't read it though. 1020