  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 $z\geq2$, 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 $z\geq2$ 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 $z\geq2$ compared to previous ASPECS measurements.
