For all classical cellular automata (CA), the transition function which computes the next state of a cell (given the current state of that cell and the states of all the cells in its neighbourhood) is primitive recursive (and usually, in fact, very simple.)
To evolve a CA form, one iteratively applies the transition function to all cells in the playfield. You couldn't technically consider this process primitive recursive, because it does not necessarily ever have to come to an end.
In fact, classical CA do not specify when their evolution should come to an end. This makes them a different kind of beast from classical Turing machines,
which have a halt state. A CA cannot directly simulate a Turing machine unless you introduce a predicate that says that you consider the CA evolution to have halted. As an example, the predicate might be that all cells in the playfield have assumed a certain state.
So when someone says a CA is Turing-complete, they're being a little flexible with the notion. In terms of computability this is generally harmless, so long as the predicate is a primitive recursive function (again, like the transition function, it usually can in fact be very simple.)
I'm not entirely sure I fully understand your final question, but it sounds like you are asking how you would prove that some other system is Turing-complete by reducing it to the Rule 110 automaton. You could do that by showing that, for every instance of your system, there is a corresponding instance of the Rule 110 automaton which faithfully represents it, and that there is some predicate that is true for Rule 110 playfields when (and only when) your system has halted.
EDIT: Everything I've said above applies to CA's in general, and I just noticed the comment regarding the initial playfield of the Rule 110 automaton.
To show that some CA's are Turing-complete, the construction (beyond stipulating a halting predicate) also involves filling the CA's playfield with some pattern beyond the trivial pattern of "all cells are in one particular state". (I believe the proof that Wireworld is Turing-complete also uses this notion; the playfield is filled with a repeating pattern of gates.) This is generally not a problem, so long as that pattern is computable. The Turing machine simulating the CA can simply write the next state in that pattern to the part of the tape that it is using to represent that part of the CA's playfield, when it comes time for the Turing machine to simulate that part of the playfield for the first time.
If you want to show that some system is Turing-complete by reducing it to a CA which has been proved Turing-complete in this manner, you would have to keep in mind that the CA's initial playfield needs to be set up like this, but beyond that it would likely not be a major consideration in your reduction.