We present ALMA CO(1-0) and CO(3-2) observations of the brightest cluster galaxy (BCG) in the 2A 0335+096 galaxy cluster (z = 0.0346). The total molecular gas mass of 1.13 ±0.15 ×109 M o is divided into two components: a nuclear region and a 7 kpc long dusty filament. The central molecular gas component accounts for 3.2 ±0.4 ×108 M o of the total supply of cold gas. Instead of forming a rotationally supported ring or disk, it is composed of two distinct, blueshifted clumps south of the nucleus and a series of low-significance redshifted clumps extending toward a nearby companion galaxy. The velocity of the redshifted clouds increases with radius to a value consistent with the companion galaxy, suggesting that an interaction between these galaxies <20 Myr ago disrupted a pre-existing molecular gas reservoir within the BCG. Most of the molecular gas, 7.8 ±0.9 ×108 M o, is located in the filament. The CO emission is co-spatial with a 104 K emission-line nebula and soft X-rays from 0.5 keV gas, indicating that the molecular gas has cooled out of the intracluster medium over a period of 25-100 Myr. The filament trails an X-ray cavity, suggesting that the gas has cooled from low-entropy gas that has been lifted out of the cluster core and become thermally unstable. We are unable to distinguish between inflow and outflow along the filament with the present data. Cloud velocities along the filament are consistent with gravitational free-fall near the plane of the sky, although their increasing blueshifts with radius are consistent with outflow.