  The heavy elements (Z>30) are created in neutron-capture processes which
happen at very different nucleosynthetic sites. To study them in an environment
different from the Milky Way, we target these elements in RGB stars in the
Sculptor dwarf spheroidal galaxy. Using ESO VLT/FLAMES spectra, we measure the
chemical abundances of Y, Ba, La, Nd, and Eu, in 98 stars covering
$-2.4<\text{[Fe/H]}<-0.9$. This is the first paper in a series about the
$n$-capture elements in dwarf galaxies, and here we focus on the relative and
absolute timescales of the slow ($s$)- and rapid ($r$)-processes in Sculptor.
From the abundances of the $s$-process element Ba and the $r$-process element
Eu, it is clear that the $r$-process enrichment occurred throughout the entire
chemical evolution history of Sculptor. Furthermore, there is no evidence for
the $r$-process to have a significant time delay relative to core-collapse
supernovae. Neutron star mergers are therefore unlikely the dominant (or only)
nucleosynthetic site of the $r$-process. However, the products of the
$s$-process only become apparent at $\text{[Fe/H]}\approx-2$ in Sculptor, and
the $s$-process becomes the dominant source of Ba at $\text{[Fe/H]}\gtrsim-2$.
We test the use of [Y/Mg] and [Ba/Mg] as chemical clocks in Sculptor. Similarly
to what is observed in the Milky Way, [Y/Mg] and [Ba/Mg] increase towards
younger ages. However, there is an offset, where the abundance ratios of [Y/Mg]
in Sculptor are significantly lower than those of the Milky Way at any given
age. This is most likely caused by metallicity dependence of yields from the
$s$-process, as well as different relative contribution of the $s$-process to
core-collapse supernovae in these galaxies. Comparisons of our data with that
of the Milky Way and the Fornax dwarf spheroidal galaxy furthermore show that
these chemical clocks are both metallicity and environment dependent.
