Can any finite problem be in NP-Complete?
If a finite problem is NP-complete then P=NP, since every finite problem has a polynomial time algorithm (even a constant time algorithm).

When we say that Sudoku is NP-complete, we mean that a generalized version of Sudoku played on an $n^2 \times n^2$ board is NP-complete.

Finally, the 4-clique problem, while not a finite problem (the input graph has unbounded size), is an easy problem which has a polynomial time algorithm.