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In this work we consider a cosmological scenario in which the Universe contracts initially having a bouncing-like behavior, and accordingly after it bounces off, it decelerates following a matter dominated like evolution and at very large positive times it undergoes through an accelerating stage. Our aim is to study such evolution in the context of $F(R)$ gravity theory, and confront quantitatively the model with the recent observations. Using several reconstruction techniques, we analytically obtain the form of $F(R)$ gravity in two extreme stages of the universe, particularly near the bounce and at the late time era respectively. With such analytic results and in addition by employing appropriate boundary conditions, we numerically solve the $F(R)$ gravitational equation to determine the form of the $F(R)$ for a wide range of values of the cosmic time. The numerically solved $F(R)$ gravity realizes an unification of certain cosmological epochs of the universe, in particular, from a non-singular bounce to a matter dominated epoch and from the matter dominated to a late time dark energy epoch. Correspondingly, the Hubble parameter and the effective equation of state parameter of the Universe are found and several qualitative features of the model are discussed. The Hubble radius goes to zero asymptotically in both sides of the bounce, which leads to the generation of the primordial curvature perturbation modes near the bouncing point. Correspondingly, we calculate the scalar and tensor perturbations power spectra near the bouncing point, and accordingly we determine the observable quantities like the spectral index of the scalar curvature perturbations, the tensor-to-scalar ratio, and as a result, we directly confront the present model with the latest Planck observations. Furthermore the $F(R)$ gravity dark energy epoch is confronted with the Sne-Ia+BAO+H(z)+CMB data.