If the same atoms (cells) that now compose my body get arranged in N years away, am I that person?
Let me start from the bottom up: If we assume the earth as a closed system, i.e. exchange of energy with the surrounding space is possible, but exchange of matter impossible, than naturally our body contains atoms previously present in other form of matter.  Already our breakfast is matter we incorporate, subject to chemical reaction (digestion), and eventually excrete in one form, or an other.  Hence it is likely that we contain some atoms that once constituted a mammoth, or a dinosaur, or by your tomato in the back yard.

Digestion often splits the larger molecules present in our food into smaller ones.  This eases transport of matter across membranes, for example.  Think of proteins de-constructed into amino acids; our organism uses the smaller to build new proteins in need.

Beside chemical degradation as in aforementioned example of the proteins, digestion includes physical degradation, too -- chewing food, for example increases the surface area the chemicals responsible for digestion may access.  By  both means of action, physical and chemical degradation, entire cells face only a tiny chance to remain intact "as such".  It is more likely that they are dismantled, are split into their constituents, eventually -- as molecules -- harvested in the ways briefly mentioned earlier.

To sum up: so far there is no evidence a synapse of yours will be recycled, maintaining the original form and function, to find its way back as synapse in an other brain, similar to a spare tire.  Atoms of yours however, as easily they entered your body, eventually will leave you; at their pace.