A polynomial of degree 3 that has three real zeros, only one of which is rational.

Yes, your answer is correct.

Another way to come up with some is to use the form $x^3-nx \;\;\forall\; n\in\Bbb N \land \sqrt n \notin \Bbb N$ (i.e. where $n$ isn't a perfect square but natural).

For the quartic, consider the equation $x^4-(a+b)x^2+(ab)\;\;\forall \;a,b \in \Bbb N \land \sqrt a, \sqrt b \notin \Bbb N$ (i.e. where $a+b$ is the sum of two non-perfect square natural numbers and $ab$ is their product).

For your question at the end, yes, a similiar technique would work.

If you know $n$ zeroes of a polynomial of degree $n$, then you know all of them, because there are at most $n$ of them. So if you know four irrational zeroes, you know that there are no rational zeroes. In particular, for a degree 4 polynomial, you know that there are no rational zeroes if you know four irrational zeroes.