Laser cooling of mechanical degrees of freedom is one of the most significant achievements in the field of optomechanics. Here, we report, for the first time to the best of our knowledge, efficient passive optomechanical cooling of the motion of a freestanding waveguide coupled to a whispering-gallery-mode (WGM) resonator. The waveguide is an 8 mm long glass-fiber nanospike, which has a fundamental flexural resonance at $\mathrm{\Omega }/2\pi =2.5\mathrm{kHz}$ and a $Q$-factor of $1.2\times {10}^5$. Upon launching $～250\mathrm{\mu W}$ laser power at an optical frequency close to the WGM resonant frequency, we observed cooling of the nanospike resonance from room temperature down to 1.8 K. Simultaneous cooling of the first higher-order mechanical mode is also observed. The strong suppression of the overall Brownian motion of the nanospike, observed as an 11.6 dB reduction in its mean square displacement, indicates strong optomechanical stabilization of linear coupling between the nanospike and the cavity mode. The cooling is caused predominantly by a combination of photothermal effects and optical forces between nanospike and WGM resonator. The results are of direct relevance in the many applications of WGM resonators, including atom physics, optomechanics, and sensing.