  We compute instanton corrections to the partition function of sine-Liouville
(SL) theory, which provides a worldsheet description of two-dimensional string
theory in a non-trivial tachyon background. We derive these corrections using a
matrix model formulation based on a chiral representation of matrix quantum
mechanics and using string theory methods. In both cases we restrict to the
leading and subleading orders in the string coupling expansion. Then the CFT
technique is used to compute two orders of the expansion in the SL perturbation
parameter $\lambda$, while the matrix model gives results which are
non-perturbative in $\lambda$. The matrix model results perfectly match those
of string theory in the small $\lambda$ expansion. We also generalize our
findings to the case of perturbation by several tachyon vertex operators
carrying different momenta, and obtain interesting analytic predictions for the
disk two-point and annulus one-point functions with ZZ boundary condition.
