We report a novel pilot project to characterise intra-night optical variability (INOV) of an extremely rare type of quasar, which has recently been caught in the act of transiting from a radio-quiet to radio-loud state, on a decadal time scale. Such rare transitions may signify a recurrence, or conceivably the first switch-on of jet activity in optically luminous quasars. The newly formed jet could well be jittery and unsteady, both in power and direction. The optically brightest among such radio-state transition candidates, the quasar J0950+5128 ($z = 0.2142$), was monitored by us with dense sampling in the R-band, during 2020-21 in 6 sessions, each lasting $ \gt $ 4 hours. This is the first attempt to characterise the INOV properties associated with this recently discovered, extremely rarely observed phenomenon of quasar radio-state transition. The non-detection of INOV in any of the 6 sessions, down to the 1-2% level, amounts to a lack of evidence for a blazar-like optical activity, $\sim$ 2 years after its transition to radio-loud state was found. The only INOV feature detected in J0950+5128 during our observational campaign was a $\sim$ 0.15-mag spike lasting < 6 minutes, seen at 13.97 UT on 18-March-2021. We also report the available optical light curves of this quasar from the Zwicky Transient Facility survey, which indicate that it had experienced a phase of INOV activity around the time its transition to the radio-loud state was detected, however that phase did not sustain until the launch of our INOV campaign $\sim$ 2 years later.