How do we calculate the effective mass of an electron in an Aluminium lattice? Is there any simple analytical way to work it out?
The effective mass can be determined from a exactly known band structure. As genneth already mentioned there is no simple analytical way to get a from first principles to the Aluminium band structure.
In practice there are a number of very successful numerical approaches to calculate band structures, e.g. Hartree-Fock, LDA+U, DMFT, Tight binding and the list goes on depending on the exact material and properties you are interested in.
If you use a number of material parameters as the electron density, Fermi energy or others you can express the effective mass in those terms but that is more a reformulation than a way to calculate it.