Context. Asteroseismic modelling of isolated stars presents significant challenges due to the difficulty in accurately determining stellar parameters, particularly the stellar age. These challenges can be overcome by observing stars in open clusters whose coeval members share an initial chemical composition. The light curves from the all-sky survey by the Transiting Exoplanet Survey Satellite (TESS) allow us to investigate and analyse stellar variations in clusters with an unprecedented level of detail for the first time. Aims. We aim to detect gravity-mode oscillations in the early-type main-sequence members of the young open cluster NGC 2516 to deduce their internal rotation rates. Methods. We selected the 301 member stars with no more than mild contamination as our sample. We analysed the full-frame image light curves, which provide nearly continuous observations in the first and third years of TESS monitoring. We also collected high-resolution spectra using the Fiber-fed Extended Range Optical Spectrograph for the g-mode pulsators, with the aim of assessing the Gaia effective temperatures and gravities and preparing for future seismic modelling. Results. By fitting the theoretical isochrones to the colour-magnitude diagram of a cluster, we determined an age of 102 ± 15 Myr and inferred that the extinction at 550 nm (A0) is 0.53 ± 0.04 mag. We identified 147 stars with surface-brightness modulations: 24 with gravity (g-)mode pulsations (γ Doradus or slowly pulsating B-type stars) and 35 with pressure (p-)mode pulsations (δ Sct stars). When sorted by colour index, the amplitude spectra of the δ Sct stars show a distinct ordering and reveal a discernible frequency-temperature relationship. The near-core rotation rates, measured from period spacing patterns in two slowly pulsating B-type (SPB) stars and nine γ Dor stars, reach up to 3 d-1. This is at the high end of the values found from Kepler data of field stars of similar variability type. The γ Dor stars of NGC 2516 have internal rotation rates as high as 50% of their critical value, whereas the SPB stars exhibit rotation rates close to their critical rate. Although the B-type stars are rotating rapidly, we did not find long-term brightness and colour variations in the mid-infrared, which suggests that there are no disc or shell formation events in our sample. We also discussed the results of our spectroscopic observations for the g-mode pulsators.