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The one-electron universe postulate, proposed by John Wheeler in a
telephone call to Richard Feynman in the spring of 1940, hypothesises
that all electrons and positrons are actually manifestations of a
single entity moving backwards and forwards in time. According to
Feynman:

“ I received a telephone call one day at the graduate college at
Princeton from Professor Wheeler, in which he said, "Feynman, I know
why all electrons have the same charge and the same mass" "Why?"
"Because, they are all the same electron!"
https://en.wikipedia.org/wiki/One-electron_universe

The supposition is that there is only one electron. It stands to reason that if I could change that electron, I could change the properties of the whole Universe.

Assume that the postulate is true.

Question

Given foreseeable science, what properties of a single electron could be changed? In theory, could we change the charge? Could we change the mass?

This universe is fundamentally impossible, since some electrons have their worldlines terminated in a black hole. Without a full working model of quantum gravity, we can't make any firm predictions about what happens to such electrons, other than the singularity is likely to end their existence. The black hole will inherit the charge, mass and angular momentum, but lose all the electron-ness of the particle's information (no hair theorem).

You can also have electrons terminated in beta capture events (which turns a proton into a neutron and the electron stops existing).

So, there won't be anything you can change, because there won't be just one electron. The model is completely incompatible with current understanding of physics.

Whether the Wheeler postulate is true or not, under the known laws of physics, you can't change any of the properties of an electron. Mass, charge, spin, magnetic moment, etc. are all intrinsic properties. They are not mutable by any known (or even to the best of my knowledge hypothesized) mechanism.

As to the postulate itself, as pointed out in comments and in tylisrn's answer, there are strong reasons to not believe that the Wheeler postulate is true. In addition to problems with any mechanism that results in the creation or destruction of a lone electron or positron, the Wheeler postulate runs into difficulties in explaining the observed imbalance of matter and anti-matter. If a single electron is zipping backwards and forwards through time, we should see it moving backwards as we see it moving forwards. This would imply equal numbers of electrons and positrons, which we simply do not observe.