We present a power spectrum analysis of the ALMA Spectroscopic Survey Large Program (ASPECS LP) data from 84 to 115 GHz. These data predominantly probe small-scale fluctuations ($k=10$-$100$ h Mpc$^{-1}$) in the aggregate CO emission in galaxies at $1 lesssim z lesssim 4$. We place an integral constraint on CO luminosity functions (LFs) in this redshift range via a direct measurement of their second moments in the three-dimensional (3D) auto-power spectrum, finding a total CO shot noise power $P_{textrm{CO,CO}}(k_{textrm{CO(2-1)}}) leq 1.9times10^2$ $mu$K$^2$ (Mpc h$^{-1}$)$^3$. This upper limit ($3sigma$) is consistent with the observed ASPECS CO LFs in Decarli et al. 2019, but rules out a large space in the range of $P_{textrm{CO,CO}}(k_{textrm{CO(2-1)}})$ inferred from these LFs, which we attribute primarily to large uncertainties in the normalization $Phi_*$ and knee $L_*$ of the Schechter-form CO LFs at $z > 2$. Also, through power spectrum analyses of ASPECS LP data with 415 positions from galaxies with available optical spectroscopic redshifts, we find that contributions to the observed mean CO intensity and shot noise power of MUSE galaxies are largely accounted for by ASPECS blind detections, though there are $sim20$% contributions to the CO(2-1) mean intensity due to sources previously undetected in the blind line search. Finally, we sum the fluxes from individual blind CO detections to yield a lower limit on the mean CO surface brightness at 99 GHz of $langle T_{textrm{CO}} rangle = 0.55pm0.02$ $mu$K, which we estimate represents $68$-$80$% of the total CO surface brightness at this frequency.
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