Can mass of an electron be measured without measuring its charge?

Yes, the mass of the electron can be measured by the energy of the gamma rays involved in electron-positron annihilation. The mass of an electron is equal to the energy (divided by the speed of light squared) of one of those gamma rays, as measured in the center of momentum frame.

The measurement of gamma ray energies is known as Gamma-ray spectroscopy. The general idea is that the gamma ray is absorbed by some material, and the resulting ionization is detected in some way (cheaper detectors tend to use a scintillating material, while more expensive detectors use semiconductor technologies). The detector response is calibrated with a source of known gamma energies, and then the energy of another gamma ray can be measured.

The detector has a finite resolution, but the measurement can be enhanced by looking at extra features at the spectrum. For instance, there's a "Compton edge" that appears from gamma rays that Compton scatter rather than being absorbed, and an extra peak at twice the electron mass (when both gamma rays from the annihilation are absorbed).

That said, the relative uncertainty on the electron mass is only around $10^{-8}$, which is much better than you can get with standard gamma ray spectroscopy.

Some physics programs (Rutger's, for instance) have their students actually run this experiment in a lab course.

One can measure the mass of the electron using diffraction of electrons.

EDIT (2/6/2018): The de Broglie wavelength only depends on the mass and velocity, so if electrons diffract, say, on a crystal or on two slits, one can determine the distance between the maxima of the diffraction pattern and calculate the mass, if the velocity of electrons is known. One does not need to know the charge of electrons to prepare a bunch of electrons with known velocity (using, for example, a thermal emission source and the time of flight method).