Diamondoids are small diamond nanocrystals with perfect hydrogenated surfaces. Recent absorption measurements showed that the spectrum of diamondoids exhibit features that are not understood from the theoretical point of view, e.g. optical gaps are only slightly larger than the gap of bulk diamond which runs against the quantum confinement effect. Previous calculations, even beyond standard density functional theory (DFT), failed to obtain the experimental optical gaps (E g) of diamondoids. We show that all-electron time-dependent DFT (TD-DFT) calculations including the PBEO hybrid functional in the TD-DFT kernel are able to provide quantitatively accurate results. Our calculations demonstrate that Rydberg transitions govern the low energy part of the absorption spectrum, even for relatively large nanodiamonds resulting in low E g. Since the optical gap of these diamondoids lies in the ultraviolet spectral region, we investigated whether simple adsorbates of the surface are able to shift the gap towards the infrared region. We found that a double bonded sulfur atom at the surface results in a substantial gap reduction.