If A is polynomial-time reducible to B and B is NP-Complete, can I say that A is NP-Complete as well?
You can indeed not guarantee that A is NP-complete.
Problem A: Given is a single number X ≥ 1. Is X equal to 1? Clearly in P, and in NP, but of course not NP-complete.
Problem B: Given n items of size $s_i$ ≥ 1, a bin size B ≥ 1, and an integer k ≥ 1, can the n items be put into k bins of size B? This one is NP complete.
A can be reduced to B in polynomial time: Just take 1 item of size X, let B = 1 and k = 1.