What equipment is needed to perform simple plant electrophysiology?
Electrophysiology on plant cells means single-cell recordings, as there are no gross compound action potentials.  As you correctly say, this means micro-electrodes; this in turn means computerized manipulation of the pipettes, a decent microscope, signal amplifier, recording software, stimulator slash voltage/current control, Faraday cage, analysis software (Matlab), and not unimportantly a vibration-free workbench (whole departments have moved house due to micrometer vibrations in the building). I guess you should think anywhere between 10k - 100k (dollars, euros whatever) as a reasonable estimate.

A student internship would be advisable finance-wise, but especially because single cell recordings, and electrophysiological skills in general, are relatively difficult to master.