The measurement of the temporal evolution in the redshift of distant objects, the redshift drift, is a probe of universal expansion and cosmology. We perform the first steps towards a measurement of such effect using the Lyman-alpha forest in the spectra of bright quasars as a tracer of cosmological expansion. Our goal is to determine to which precision a velocity shift measurement can be carried out with the signal-to-noise (S/N) level currently available and whether this precision aligns with previous theoretical expectations. A precise assessment of the achievable measurement precision is fundamental for estimating the time required to carry out the whole project. We acquire 12 hours of ESPRESSO observations distributed over 0.875 years of the brightest quasar known, J052915.80-435152.0 (z=3.962), to obtain high-resolution spectra of the Lyman-alpha forest, with median S/N of ~86 per 1km/s pixel at the continuum. We divide the observations into two epochs and analyse them using both a pixel-by-pixel method and a model-based approach. This comparison allows us to estimate the velocity shift between the epochs, as well as the velocity precision that can be achieved at this S/N. The model-based method is calibrated using high-resolution simulations of the intergalactic medium, and it provides greater accuracy compared to the pixel-by-pixel approach. We measure a velocity drift of the Lyman-alpha forest consistent with zero: {DELTA}v=-1.25+/-4.45m/s, equivalent to a cosmological drift of dv/dt=-1.43+/-5.09m/s/yr or dz/dt=(-2.19+/-7.77)x10^-8^yr^-1^. The measurement uncertainties are on par with the expected precision. We estimate that reaching a 99% detection of the cosmic drift requires a monitoring campaign of 5400 hours of integration time over 54 years with an ELT and an ANDES-like high-resolution spectrograph.