We present a strong lensing analysis of the galaxy cluster SDSS J1029+2623 at z = 0.588, one of the few currently known lens clusters with multiple images of a background (z = 2.1992) quasar with a measured time delay. We use archival Hubble Space Telescope multiband imaging and new Multi Unit Spectroscopic Explorer follow-up spectroscopy to build an accurate lens mass model, a crucial step toward future cosmological applications. The spectroscopic data enable the secure identification of 57 cluster members and of two nearby perturbers along the line of sight. We estimate the inner kinematics of a subset of 20 cluster galaxies to calibrate the scaling relations parameterizing the sub-halo mass component. We also reliably determine the redshift of four multiply imaged sources, provide a tentative measurement for one system, and report the discovery of a new four-image system. The final catalog comprises 26 multiple images from seven background sources, spanning a wide redshift range, from 1.02 to 5.06. We present two parametric lens models, with slightly different cluster mass parameterizations. The observed positions of the multiple images are accurately reproduced within approximately 0.″2, the three image positions of the quasar within only ∼ 0.″1. We estimate a cluster projected total mass of M(<300 kpc) ∼2.1 × 1014 M ⊙, with a statistical uncertainty of a few percent. Both models, which include a small galaxy close to one of the quasar images, predict magnitude differences and time delays between the quasar images that are consistent with the observations.