It is stated in our lecture notes without proof that the center of $SU(2)=\{\pm 1\}$. I understand how to find the center of $SO(3)$, which is $\{1\}$ and that is given in the notes, is that somehow useful in finding $Z(SU(2))$?
Note: $SU(2)$ is the special unitary group of degree two, meaning $2\times 2$ matrices with determinant$=+1$. https://en.wikipedia.org/wiki/Special_unitary_group
We use the following characterization of $SU(2)$:
$$SU(2) = \left\{A= \pmatrix{\alpha & -\bar \beta \\ \beta & \bar\alpha}: \alpha, \beta\in \mathbb C,\ |\alpha|^2+|\beta|^2 = 1\right\}$$
So if $A \in SU(2)$ lie in the center, then $AB = BA$ for all $B\in SU(2)$. In particular it is true for
$$B= \pmatrix{0 & 1 \\ -1 & 0},\ \ \pmatrix{0 & i \\ i & 0 }\in SU(2)$$
Using the first matrix, one has that $\alpha$ is real and $\beta$ is imaginary. Using the second matrix, you have that $\beta = 0$. Thus $A$ is diagonal with real entries and the only two matrices in $SU(2)$ with this properties are $\pm I$.