  We report on the first multi-wavelength Swift monitoring campaign performed
on SDSS J164100.10+345452.7, a nearby narrow-line Seyfert 1 galaxy formerly
known as radio quiet which was recently detected both in the radio (at 37 GHz)
and in the $\gamma$-rays, which hints at the presence of a relativistic jet.
During our 20-month Swift campaign, while pursuing the primary goal of
assessing the baseline optical/UV and X-ray properties of J1641, we caught two
radio flaring episodes, one each year. Our strictly simultaneous
multi-wavelength data closely match the radio flare and allow us to
unambiguously link the jetted radio emission of J1641. Indeed, for the X-ray
spectra preceding and following the radio flare a simple absorbed power-law
model is not an adequate description, and an extra absorption component is
required. The average spectrum of J1641 can be best described by an absorbed
power law model with a photon index $\Gamma=1.93\pm0.12$, modified by a
partially covering neutral absorber with a covering fraction
$f=0.91_{-0.03}^{+0.02}$. On the contrary, the X-ray spectrum closest to the
radio flare does not require such extra absorber and is much harder
($\Gamma_{\rm flare} \sim 0.7\pm0.4$), thus implying the emergence of a
further, harder spectral component. We interpret this as the jet emission
emerging from a gap in the absorber. The fractional variability we derive in
the optical/UV and X-ray bands are found to be lower than the typical values
reported in the literature, since our observations of J1641 are dominated by
the source being in a low state. Under the assumption that the origin of the 37
GHz radio flare is the emergence of a jet from an obscuring screen also
observed in the X-rays, the derived total jet power is $P^{\rm tot}_{\rm
jet}=3.5\times10^{42}$ erg s$^{-1}$, comparable to the lowest measured in the
literature. [Abridged]
