  In this work we calculate the transition form factors of $\Lambda_b$ decaying
into proton and $N^*(1535)$ ($J^P={1/2}^+$ and ${1/2}^-$ respectively), within
the framework of light-cone sum rules with the distribution amplitudes (DAs) of
$\Lambda_b$-baryon. In the hadronic representation of the correlation function,
we have isolated both the proton and the $N^*(1535)$ states so that the
$\Lambda_b \rightarrow p, N^*(1535)$ form factors can be evaluated
simultaneously. Due to the less known properties of baryons, we investigate
three interpolating currents of the light baryons and five parametrization
models for DAs of $\Lambda_b $. Numerically, our predictions on the $\Lambda_b
\rightarrow p$ form factors and the branching fractions of $\Lambda_b\to p\ell
\nu$ from the Ioffe or the tensor currents are consistent with the Lattice
simulation and the results from the light-baryon sum rules, as well as the
experimental data of $Br(\Lambda_b^0\to p\mu^-\bar\nu)$. The predictions on the
form factors of $\Lambda_b \rightarrow N^*(1535)$ are very sensitive to the
choice of the interpolating currents so that the relevant measurement could be
helpful to clarify the properties of baryons.
