Context. XMMU J1229+0151 is a rich galaxy cluster with redshift $z=0.975$ that was serendipitously detected in X-rays within the scope of the XMM-Newton Distant Cluster Project. Both HST/ACS observations in the i775 and z850 passbands and VLT/FORS2 spectroscopy were obtained, in addition to follow-up Near-Infrared (NIR) imaging in the J- and Ks-bands with NTT/SOFI. Aims. We investigate the photometric, structural, and spectral properties of the early-type galaxies in the high-redshift cluster XMMU J1229+0151. Methods. Source detection and aperture photometry are performed in the optical and NIR imaging. Galaxy morphology is inspected visually and by means of Sersic profile fitting to the 21 spectroscopically confirmed cluster members in the ACS field of view. The $i_{775}-z_{850}$ colour-magnitude relation (CMR) is derived with a method based on galaxy magnitudes obtained by fitting the surface brightness of the galaxies with Sersic models. Stellar masses and formation ages of the cluster galaxies are derived by fitting the observed spectral energy distributions (SED) with models developed by Bruzual & Charlot. Star-formation histories of the early-type galaxies are constrained by analysing the stacked spectrophotometric data. Results. The structural Sersic index n obtained by model fitting agrees with the visual morphological classification of the confirmed members, indicating a clear predominance of elliptical galaxies (15/21). The $i_{775}-z_{850}$ colour-magnitude relation of the spectroscopic members shows a very tight red-sequence with a zero point of 0.86 ± 0.04 mag, and intrinsic scatter equal to 0.039 mag. The CMR obtained with the galaxy models has similar parameters. By fitting both the spectra and SED of the early-type population, we obtain a star-formation-weighted age of 4.3 Gyr for a median galaxy stellar-mass of 7.4 $\times$$10^{10}~M_{odot}$. Instead of identifying a brightest cluster galaxy (BCG) unambiguously, we find three bright galaxies with a similar z850 magnitude, which are, in addition, the most massive cluster members, with ~2 $\times$$10^{11}~M_{odot}$. Our results strengthen the current evidence of a lack of significant evolution in both the scatter and slope of the red-sequence out to z $\thicksim$ 1.
