There is an adjunction $$ (\pi_0 \dashv disc): Spaces \rightarrow Set $$ where $\pi_0$ sends a space to its path components, and $disc$ sends a set to the space with discrete topology.
(i) Does $\pi_0$ preserve pullback? If not what is a counter example.
I feel this is true. But i've never heard of some saying it preserves finite limits. (We know it preserves products)
Proof: Proof $A\rightarrow B \leftarrow A'$, a map from interval $I$ to the fiber product is identified as the set $\pi_0(A) \times_{\pi_0(B) } \pi_0(A')$ by universal property.
ii) Does $\pi_0$ preserve effective epimorphism?
If (i) is true, then as $\pi_0$ commutes with fiber products and colimits it naturally preserves effective epis.
No, $\pi_0$ preserves neither pullbacks nor effective epimorphisms. For pullbacks, for instance, consider two maps $f,g:[0,1]\to S^1$ which trace out arcs which are disjoint except at their endpoints. The pullback is then $\{(0,0),(1,1)\}$ which is disconnected, but $[0,1]$ and $S^1$ are both path-connected so the pullback after applying $\pi_0$ is just a single point.
For effective epimorphisms, let $X$ be the (closed) topologist's sine curve and let $p:X\to Y$ be the quotient map that collapses the line segment where the sine curve accumulates to a point. Then $p$ is an effective epimorphism, but $\pi_0(X)$ has two points while $\pi_0(Y)$ has one. See this answer for more details and related discussion.