The origin and evolution of the X-ray emission in very young stellar objects (YSOs) are not yet well understood because it is very hard to observe YSOs in the protostellar phase. Using COUP data, we study the X-ray properties of stars in the ONC in different evolutionary classes: luminosities, hydrogen column densities N_H, effective plasma temperatures, and time variability are compared to understand if the interaction between the circumstellar material and the central object can influence the X-ray emission. We have assembled the deepest and most complete photometric catalog of objects in the ONC region from the UV to 8 μm using data from the HST Treasury Program; deep and almost simultaneous UBVI and JHK images taken, respectively, with WFI at ESO 2.2 m and ISPI at CTIO 4 m telescopes; and Spitzer IRAC imaging. We select high-probability candidate Class 0-I protostars, a sample of "bona fide" Class II stars, and a set of Class III stars with IR emission consistent with normal photospheres. Our principal result is that Class 0-Ia objects are significantly less luminous in X-rays, in both the total and hard bands, than the more evolved Class II stars with mass larger than 0.5 M_⊙ the latter show X-ray luminosities similar to those of Class 0-Ib stars. This result supports the hypothesis that the onset of X-ray emission occurs at a very early stage of star formation. Spectral properties of Class 0-I stars are similar to those of the more evolved Class II and III objects, except for a larger absorption likely due to gas in the envelope or disk of the protostellar objects. Our data suggest that the three different classes have similar X-ray temporal variability.