How does the energy stored in the electric field of the electron relates to its rest mass?
It depends on whether we assume the electron has finite charge density everywhere or not.
In case the charge density of electron is finite everywhere (like it is in the Lorentz and Abraham models of the electron, where charge is distributed on the surface or throughout the volume of a sphere), Poynting's equation is valid everywhere and implies expression for EM energy density you wrote above. It can be shown that net result of mutual EM forces between parts of the sphere results in increase of effective rest mass and other effects, like radiation damping. The change in the rest mass can be then related to Poynting's energy of the electron's field. However, how large these effects are depends on many details, like the size of the sphere, distribution of charge in it and nature of non-EM forces that hold the electric charge together. It is possible that change in the mass is very small part of total mass, but it could also be a substantial part.
In case the electron's charge is concentrated at some point so density is infinite, local Poynting's equation is invalid at that point and thus cannot be relied upon to calculate total EM energy. For example, if the electrons are points, one needs to use theory of point particles to calculate their EM energy. In this kind of theory, a theorem analogous to Poynting's can be derived. It implies different formula for EM energy density where one charged point particle has EM field, but there is zero EM energy associated with it. Only if there are several particles, the net EM energy can be non-zero.
For example, in Frenkel-type theory of electrons, the electrons are points with individual EM fields. The particles interact via EM forces but one electron has no parts that could interact among themselves, so there is no change in its mass due to EM interactions. Also there is no EM energy associated with the EM field of one lone electron.
How large part of the rest mass comes from this field?
We do not know if electron is extended or point-like. Consequently, we do not know what part of its mass if any can be related to EM energy stored in the space around it. In the end of the 19th century and first years of 20th century there was an hypothesis that all mass of the electron is electromagnetic mass and Kaufmann's experiments on the behaviour of fast electrons in electric and magnetic field seemed to support it. This idea was largely abandoned when special relativity got accepted, because in special relativity electromagnetic and non-electromagnetic mass behave the same. The past experiments got reinterpreted in such a way that no evidence of EM mass could be found from them.
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. http://dx.doi.org/10.1007/BF01331692
J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433. http://dx.doi.org/10.1103/RevModPhys.21.425