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How many distinct homomorphisms $\phi$ $:$ $C$$_3$ $\rightarrow$ $C$$_1$$_2$ are there? How many of these $\phi$ are ring homomorphisms $\Bbb Z$$_3$ $\rightarrow$ $\Bbb Z$$_1$$_2$?

Let $C$$_3$ $=$ $$ and $C$$_1$$_2$ $=$ $$. Then assignment $\rho$($x$) $=$ $y$$^k$ extends to a group homomorphism $\rho$($x$$^m$) $=$ $y$$^m$$^k$ iff $y$$^3$$^k$ $=$ $1$ iff $12$$\vert$$3k$ iff $4$$\vert$$k$. Thus, there are 3 distinct homomorphisms ($k$ $=$ $0,4,8$). This completes the first part.

How do I find the second part?

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Hint: a ring homomorphism must send $1$ to $1$.

If your class$^*$ doesn't require ring homomorphisms to preserve the multiplicative identity, then at a minimum, $1$ must be sent to an idempotent $e$, i.e., $e^2=e$. Use the Chinese Remainder Theorem to break down $\mathbb Z_{12}$.

$*$ Or whatever context you might be studying this in.

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    Not unless explicitly stated. Some authors are satisfied as long as $1$ is sent to an idempotent element. That being said, there are only two idempotent elements in $\Bbb Z_{12}$. – 2017-02-07
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    @Arthur I've kind of been under the impression that, culturally speaking, all ring homomorphisms are unital unless otherwise stated. Do you think I should delete this answer? – 2017-02-07
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    Never mind, I edited accordingly. – 2017-02-07