Ap Librae is one out of a handful of low-frequency peaked blazars to be detected at TeV γ-rays and the only one with an identified X-ray jet. Combined observations of Fermi-LAT at high energies (HE) and of H.E.S.S. at very high energies (VHE) revealed a striking spectral property of Ap Librae; the presence of a broad high-energy component that extends more than nine orders of magnitude in energy and is, therefore, hard to be explained by the usual single-zone synchrotron self-Compton model. We show that the superposition of different emission components related to photohadronic interactions can explain the γ-ray emission of Ap Librae without invoking external radiation fields. We present two indicative model fits to the spectral energy distribution of Ap Librae where the VHE emission is assumed to originate from a compact, sub-pc scale region of the jet. A robust prediction of our model is VHE flux variability on time-scales similar to those observed at X-rays and HE γ-rays, which can be further used to distinguish between a sub-pc or kpc scale origin of the TeV emission. We thus calculate the expected variability signatures at X-rays, HE and VHE γ-rays and show that quasi-simultaneous flares are expected, with larger amplitude flares appearing at γ-rays. We assess the detectability of VHE variability from Ap Librae with CTA, next generation of IACT. We show that ∼h time-scale variability at Eγ > 0.1 TeV could be detectable at high significance with shorter exposure times than current Cherenkov telescopes.