Nontrivial momentum-space spin texture of electrons can be induced by spin-orbit coupling and underpins various spin transport phenomena, such as current-induced spin polarization and the spin Hall effect. In this work, we find a nontrivial spin texture, spin antivortex, can appear at certain momenta on the Γ-K line in a 2D monolayer Pb on top of SiC. Different from spin vortex due to the band degeneracy in the Rashba model, the existence of this spin antivortex is guaranteed by the Poincaré-Hopf theorem and thus topologically stable. Accompanied with this spin antivortex, a Lifshitz transition of Fermi surfaces occurs at certain momenta on the K-M line, and both phenomena are originated from the anticrossing between the j=1/2 and j=3/2 bands. A rapid variation of the response coefficients for both the current-induced spin polarization and spin Hall conductivity is found when the Fermi energy is tuned around the spin antivortex. Our work demonstrates the monolayer Pb as a potentially appealing platform for spintronic applications.