We investigate the CO excitation and interstellar medium (ISM) conditions in a cold gas mass-selected sample of 22 star-forming galaxies at $z=0.46-3.60$, observed as part of the ALMA Spectroscopic Survey in the Hubble Ultra Deep Field (ASPECS). Combined with VLA follow-up observations, we detect a total of 34 CO $J rightarrow J-1$ transitions with $J=1$ up to $8$ (and an additional 21 upper limits, up to $J=10$) and six [C I] ${^3P}_1 rightarrow {^3P}_0$ and ${^3P}_2 rightarrow {^3P}_1$ transitions (and 12 upper limits). The CO(2-1) and CO(3-2)-selected galaxies, at $z=1.2$ and $2.5$, respectively, exhibit a range in excitation in their mid-$J=4,5$ and high-$J=7,8$ lines, on average lower than ($L_{rm IR}$-brighter) BzK-color- and submillimeter-selected galaxies at similar redshifts. The former implies that a warm ISM component is not necessarily prevalent in gas mass-selected galaxies at $z=1.2$. We use stacking and Large Velocity Gradient models to measure and predict the average CO ladders at $z<2$ and $zgeq2$, finding $r_{21}=0.75 pm 0.11$ and $r_{31}=0.77 pm 0.14$, respectively. From the models, we infer that the galaxies at $zgeq2$ have intrinsically higher excitation than those at $z<2$. This fits a picture in which the global excitation is driven by an increase in the star formation rate surface density of galaxies with redshift. We derive a neutral atomic carbon abundance of $(1.9 pm 0.4) times 10^{-5}$, comparable to the Milky Way and main-sequence galaxies at similar redshifts, and fairly high densities ($geq 10^4$ cm$^{-3}$), consistent with the low-$J$ CO excitation. Our results imply a decrease in the cosmic molecular gas mass density at $zgeq2$ compared to previous ASPECS measurements.
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