The neutral atom doesn't attract nor repel an additional electron when it's far.
When the second electron is near the proton, it gets attracted too, while the electrons repel an other. The trick is that the electrons arrange among themselves and, as a mean, are farther apart than their distance to the proton.
Note 1: the ability of electrons to arrange themselves holds for more electrons in a bigger atom.
Note 2: the orbitals are "stationary", that is, immobile, static, and so on. So it's "if" one electron is here and the other is there, not "when" as many sources write. ψ(r1, r2, t) has a magnitude independent of t but differs from any attempted ψ(r1, t)×ψ(r2, t) to reflect this.