  The $\beta^-$ decay $Q$-value of $^{136}$Cs ($J^\pi = 5^+$, $t_{1/2} \approx
13$~days) was measured with the JYFLTRAP Penning trap setup at the Ion Guide
Isotope Separator On-Line (IGISOL) facility of the University of Jyv\"askyl\"a,
Finland. The mono-isotopic samples required in the measurements were prepared
with a new scheme utilised for the cleaning, based on the coupling of dipolar
excitation with Ramsey's method of time-separated oscillatory fields and the
phase-imaging ion-cyclotron-resonance (PI-ICR) technique. The $Q$ value is
determined to be 2536.83(45) keV, which is $\sim$4 times more precise and
11.4(20) keV ($\sim$ 6$\sigma$) smaller than the adopted value in the most
recent Atomic Mass Evaluation AME2020. The daughter, $^{136}$Ba, has a 4$^+$
state at 2544.481(24) keV and a $3^-$ state at 2532.653(23) keV, both of which
can potentially be ultralow $Q$-value end-states for the $^{136}$Cs decay. With
our new ground-to-ground state $Q$ value, the decay energies to these two
states become -7.65(45) keV and 4.18(45) keV, respectively. The former is
confirmed to be negative at the level of $\sim$ 17$\sigma$, which verifies that
this transition is not a suitable candidate for neutrino mass determination. On
the other hand, the slightly negative $Q$ value makes this transition an
interesting candidate for the study of virtual $\beta$-$\gamma$ transitions.
The decay to the 3$^{-}$ state is validated to have a positive low $Q$ value
which makes it a viable candidate for neutrino mass determination. For this
transition, we obtained a shell-model-based half-life estimate of
$2.1_{-0.8}^{+1.6}\times10^{12}$ yr.
