Running current into a guitar pickup
Would it be possible to do the reverse of this, and feed an alternating current into a pickup to manually oscillate the string?


Yes. If we were to think of guitar pickups in this way normally, we'd call them electromagnets. To the extent that you run a current through the pickup, it will attract the string magnetically.


  excite the strings, with a frequency that I could control with a micro controller


That might not work as you expect. To get a tone out of the strings, you need the strings to resonate with the frequency you are using to excite them. The string's resonant frequency is determined by its mass and length. Without your fingers to change the length of the string, you are limited to the pitch of open strings.

If you try to excite the strings at other frequencies, you might get a sound, but it probably won't sound like a natural guitar. Whatever sound you do get will be colored by the resonances of the strings. You might get some interesting effects at harmonic frequencies. At other frequencies it may just sound very dull and quiet, or not be audible at all.