Spectroscopy for 247 stars towards the young cluster NGC 346 in the Small Magellanic Cloud has been combined with that for 116 targets from the VLT-FLAMES Survey of Massive Stars. Spectral classification yields a sample of 47 O-type and 287 B-type spectra, while radial-velocity variations and/or spectral multiplicity have been used to identify 45 candidate single-lined (SB1) systems, 17 double-lined (SB2) systems, and one triple-lined (SB3) system. Atmospheric parameters (Teff and log g) and projected rotational velocities (ve sin i) have been estimated using TLUSTY model atmospheres; independent estimates of ve sin i were also obtained using a Fourier Transform method. Luminosities have been inferred from stellar apparent magnitudes and used in conjunction with the Teff and ve sin i estimates to constrain stellar masses and ages using the BONNSAI package. We find that targets towards the inner region of NGC 346 have higher median masses and projected rotational velocities, together with smaller median ages than the rest of the sample. There appears to be a population of very young targets with ages of less than 2 Myr, which have presumably all formed within the cluster. The more massive targets are found to have lower projected rotational velocities consistent with previous studies. No significant evidence is found for differences with metallicity in the stellar rotational velocities of early-type stars, although the targets in the Small Magellanic Cloud may rotate faster than those in young Galactic clusters. The rotational velocity distribution for single non-supergiant B-type stars is inferred and implies that a significant number have low rotational velocity (≃10% with ve < 40 km s-1), together with a peak in the probability distribution at ve≃ 300 km s-1. Larger projected rotational velocity estimates have been found for our Be-type sample and imply that most have rotational velocities between 200-450 km s-1. Based on observations at the European Southern Observatory in programmes 171.D-0237 and 074D.0011.