  We address the teleportation of single- and two-qubit quantum states,
parametrized by weight $\theta$ and phase $\phi$ parameters, in the presence of
the Unruh effect experienced by a mode of a free Dirac field. We investigate
the effects of the partial measurement (PM) and partial measurement reversal
(PMR) on the quantum resources (QRs) and quantum Fisher information (QFI) of
the teleported states. In particular, we discuss the optimal behavior of the
QFI, quantum coherence (QC) as well as fidelity with respect to the PM and PMR
strength and examine the effect of the Unruh noise on optimal estimation. It is
found that in the single-qubit scenario, the PM (PMR) strength at which the
optimal estimation of the phase parameter occurs, is the same as the PM (PMR)
strength with which the teleportation fidelity and the QC of the teleported
single-qubit state reaches to its maximum value. On the other hand,
generalizing the results to two-qubit teleportation, we find that the encoded
information in the weight parameter is better protected against the Unruh noise
in two-qubit teleportation than the one-qubit scenario. However, extraction of
information encoded in the phase parameter is more efficient in single-qubit
teleportation than the two-qubit one.
