There are a few facts to use, here.
Fact 1: Entrywise complex conjugation of matrices respects addition and multiplication of matrices. That is, $\overline{A+B}=\overline{A}+\overline{B}$ and $\overline{C\cdot D}=\overline{C}\cdot\overline{D}$ for any compatibly-dimensioned matrices $A,B,C,D$.
Fact 2: $(A+B)^t=A^t+B^t$ and $(CD)^t=D^tC^t$ for any compatibly-dimensioned matrices $A,B,C,D$.
Fact 3: Skew-hermitian matrices are diagonalizable.
Given that $A,X$ are skew-hermitian $2\times 2$ matrices, how can we use the first two facts to rewrite $\overline{\text{ad}_A(X)}\,{}^t$? That should let us use the third fact to get the desired conclusion.
As for your particular example, remember that your eigenvectors will be matrices. You could look for $4$ linearly independent skew-hermitian eigenvectors of $\text{ad}_A$, but that might be kind of obnoxious, and we're really only interested in the eigenvalues, anyway.
I recommend you start with a general matrix $X=\left[\begin{array}{cc}a+bi&c+di\\u+vi&x+yi\end{array}\right].$ Now, for $X$ to be skew-hermitian, it is necessary and sufficient that $a=x=0,$ $v=d$, and $u=-c.$ (Why?) Thus, a general skew-hermitian matrix $X$ has the form $X=\left[\begin{array}{cc}bi&c+di\\-c+di&yi\end{array}\right].$ Since there are $4$ free real parameters, then $M$ has dimension at most $4$ as a vector space over $\Bbb R$. In particular, letting $V_1=\left[\begin{array}{cc}i&0\\0&0\end{array}\right],\quad V_2=\left[\begin{array}{cc}0&1\\-1&0\end{array}\right],\quad V_3=\left[\begin{array}{cc}0&i\\i&0\end{array}\right],\quad V_4=\left[\begin{array}{cc}0&0\\0&i\end{array}\right],$ we find that $M$ has $\{V_1,V_2,V_3,V_4\}$ as a basis. Our general $2\times2$ skew-hermitian matrix can then be uniquely written in the form $X=bV_1+cV_2+dV_3+yV_4$, and it can be determined through calculation that with $X$ as above, $\text{ad}_A(X)=2dV_1+(y-b)V_3-2dV_4.$ Consider then the equivalent linear transformation $T:\Bbb R^4\to\Bbb R^4$ given by $T(b,c,d,y)=(2d,0,y-b,-2d)$. It will suffice to find the eigenvalues of $T$.