  Sub-mm surveys toward overdense regions in the early Universe are essential
to uncover the obscured star-formation and the cold gas content of assembling
galaxies within massive dark matter halos. In this work, we present deep ALMA
mosaic observations covering an area of $\sim 2'\times2'$ around MQN01 (MUSE
Quasar Nebula 01), one of the largest and brightest Ly-$\alpha$ emitting
nebulae discovered thus far surrounding a radio-quiet quasar at $z\simeq3.25$.
Our observations target the 1.2- and the 3-mm dust continuum, as well as the
carbon monoxide CO(4-3) transition in galaxies in the vicinity of the quasar.
We identify a robust sample of eleven CO line-emitting galaxies (including a
closely-separated quasar companion) which lie within $\pm 4000\,{\rm
km\,s^{-1}}$ relatively to the quasar systemic redshift. A fraction of these
objects are missed in previous deep rest-frame optical/UV surveys thus
highlighting the critical role of (sub-)mm imaging. We also detect a total of
eleven sources revealed in their 1.2-mm dust continuum with six of them having
either high-fidelity spectroscopic redshift information from rest-frame UV
metal absorptions, or CO line which place them in the same narrow redshift
range. A comparison of the CO luminosity function (LF) and 1.2-mm number count
density with that of the general fields points to a galaxy overdensity of
$\delta > 10$. We find evidence of a systematic flattening at the bright-end of
the CO LF with respect to the trend measured in blank fields. Our findings
reveal that galaxies in dense regions at $z\sim3$ are more massive and
significantly richer in molecular gas than galaxies in fields, hence enabling a
faster and accelerated assembly. This is the first of a series of studies to
characterize one of the densest regions of the Universe found so far at $z >
3$.
