Context. Transit detection from space using ultra-precise photometry led to the first detection of super-Earths with solid surfaces: CoRot-7b and Kepler-10b. Because they lie only a few stellar radii from their host stars, these two rocky planets are expected to be extremely hot.Aims. Assuming that these planets are in a synchronous rotation state and receive strong stellar winds and fluxes, previous studies have suggested that they must be atmosphere-free and that a lava ocean is present on their hot dayside. In this article, we use several dedicated thermal models of the irradiated planet to study how observations with NIRSPEC on the James Webb Space Telescope (JWST) could further confirm and constrain, or reject the atmosphere-free lava ocean planet model for very hot super-Earths.Methods. Using CoRoT-7b as a working case, we explore the consequences on the phase-curve of a non tidal-locked rotation, with the presence/absence of an atmosphere, and for different values of the surface albedo. We then simulate future observations of the reflected light and thermal emission from CoRoT-7b with NIRSPEC-JWST and look for detectable signatures, such as time lag, of those peculiarities. We also study the possibility to retrieve the latitudinal surface temperature distribution from the observed SED.Results. We demonstrate that we should be able to constrain several parameters after observations of two orbits (42 h) thanks to the broad range of wavelengths accessible with JWST: i) the Bond albedo is retrieved to within ±0.03 in most cases. ii) The lag effect allows us to retrieve the rotation period within 3 h of a non phase-locked planet, whose rotation would be half the orbital period; for longer period, the accuracy is reduced. iii) Any spin period shorter than a limit in the range 30–800 h, depending on the thickness of the thermal layer in the soil, would be detected. iv) The presence of a thick gray atmosphere with a pressure of one bar, and a specific opacity higher than 10-5 m-2 kg-1 is detectable. v) With spectra up to 4.5 μm, the latitudinal temperature profile can be retrieved to within 30 K with a risk of a totally wrong solution in 5% of the cases. This last result is obtained for a signal-to-noise ratio around 5 per resel, which should be reached on Corot-7 after a total exposure time of ~70 h with NIRSPEC and only three hours on a V = 8 star.Conclusions. We conclude that it should thus be possible to distinguish the reference situation of a lava ocean with phase-locking and no atmosphere from other cases. In addition, obtaining the surface temperature map and the albedo brings important constraints on the nature or the physical state of the soil of hot super-Earths.