Which one can be accompanied by linear filters?
TL;DR: Time shift in doable with LTI systems. Subtracting $1$ is non linear.
In discrete systems, integer shifts can be implemented with filters that are zero everywhere, except at the shift location.  This can be implemented via circulant matrices, akin to convolution. This can be made compatible with Operation 1 (taking border effects into accounts).
The second operation adds one to every value. This is not linear (but affine), and therefore does not match with linear filters.