Layered amorphous-crystalline TiNiCu alloy ribbons produced by ultrarapid quenching from the liquid state (melt spinning technique) show the two-way shape memory effect without additional processing, which makes them applicable to various micromechanical devices (microtweezers) for gripping and manipulating microobjects. The present work is devoted to the study of the influence of the rejuvenation process (cryogenic thermal cycling) and the thickness of the crystalline layer on the structure and functional properties of quasi-binary TiNi-TiCu alloy with the copper content 25 at %. It is shown that thickening of the crystalline layer significantly increases not only the enthalpy of martensitic transformation but also its critical temperatures and affects the alloy crystallization pattern and temperatures. Rejuvenation treatment transforms the interface between the amorphous and crystalline layers and changes the ratio between the B19 martensitic phase and the residual B2 austenitic phase in the martensitic state, which affects the martensitic transformation parameters. In addition, cryothermal treatment causes a noticeable increase in reversible strain (magnitude of the two-way shape memory effect) and significantly narrows the temperature hysteresis of the shape change, which can improve the functional properties of microdevices based on rapidly quenched amorphous-crystalline ribbons.